Stimson,

Since you don't live in OF world, what is your reason for trying to blend these two otherwise distinct ceremonies?

What I would suggest is this:

Ten minutes before Mass is to start, a priest (perhaps the celebrant, perhaps a priest sitting in choir) wearing Cassock and white stole, accompanied, if needed, by a server with the Holy Water, performs the ceremony of the Churching, after which the woman/women return the their places, and the priest and his server retire to the sacristy to prepare for Mass.

Then the Mass of the Purification with the Blessing of Candles takes place as usual at the appointed time.

As far as a schedule goes, I would print something like

5:20 pm: Blessing of women after childbirth.
5:30 pm: Mass of the Purification of the Blessed Virgin Mary.

Maybe hokey, but it also sounds unfamiliar. I wonder how many folks have seen anything like this in the last 50 years? What is the point of it all? Curious, not criticizing.

Who would be receiving it? It’s for mothers who have just given birth and are coming back into the Church. That’s why it’s done the day of the baby’s baptism. To do otherwise would be treating the ceremony as a novelty.

And, rather interestingly:

in the Roman Catholic Church, it is only found in the extraordinary form of the Roman Rite and in Anglican Ordinariate parishes.

It may not be widely used in the OF, but there is an official ritual, see https://www.fisheaters.com/churchingofwomen.html for English and Latin. Unfortunately it has dropped both Sub tuum and Magnificat.

This site, I love it, you might find useful too.

Of course somewhat trickier to read than @tomjaw ‘s beautifully presented typeset version but you can find a variety of variations here that may be helpful to use either to help find the errors, to access a brief modern notation summary and some very old notations too.
I’ll attach a few I got to work which may help you to fix any errors in the typeset version.

http://cantus.uwaterloo.ca/id/003047
http://www.e-codices.unifr.ch/en/fcc/0002/239r
http://www.e-codices.unifr.ch/en/csg/0391/184 St. Gallan notation here.
These may not be useful I dunno but they'll certainly be complementary to tomjaws copy if you need another point of reference.

My wife has had this blessing 7 times, it is normal in our parish for the mother to have this blessing after the baptism of her child. We have over 10 mothers in our parish that have had this blessing.

What is the problem with having a blessing for a mother after childbirth?

Anyway here is the text, second blessing.
https://sanctamissa.org/en/resources/books-1962/rituale-romanum/49-blessings-of-persons.html

Charles,

Surely, if the purification was good enough for Our Lady, modern mothers shouldn't sneeze at it, and modern husbands/fathers should want a special blessing for their wives.

You would be surprised what modern people have been re-discovering. The Extraordinary form of the Roman Rite, for example, and...… Gregorian Chant and … Polyphony.

How many around here are, say, under 35?

The link that Tomjaw provided includes precatory explanations to counter negative associations with the ritual.

People may not recall the days when mothers were not present for the baptism of their infants, even occurring a month after birth. My mother was not present for the baptism of any of her six live birthed children (her first miscarriage preceding them being an exception, when her sister tried to baptize the miscarried baby conditionally). I remember her explaining not only the ritual of churching of women but also how common it was for not only people but clerics to give it a certain negative purifying association, which even over 50 years ago appalled my mother, who often made it her mission to confront clerics who had a problem with women, as it were (and oh yes they most certainly existed); I suspect she learned this from her mother, who was not one to ever suffer foolish people in power gladly.

The BVM could and did do many things. But surely no one regarded her as ritually unclean after having given birth - she was after all still a virgin! (Howsoever that happened.)

Women I have met who were not immaculately conceived understand that the church understood them to be unclean, and in need of purification before they could take part in community life again. This may not have been what the official words said - but this was the clear meaning that they and their families took from the event.

A blessing (prayer of strengthening for the journey) for both parents at the time of baptism makes a lot of sense: they've made some weighty commitments, and need support in them. But neither parent is (necessarily) unclean, or should be treated in that way.

I never understood women to be 'ritually unclean' after birth. Did you seriously think I did?

Ah this is a tricky question @charlesw and @paxmelodius and one I needed to research a little myself. The first time I discovered that Our Lady took around to a temple some turtle doves in order to become purified I laughed. Purified from what? Can’t be sex, can’t be sin she didn’t have either of those things. So, why did she go?

This has less to do with virginity and more to do with pain, blood, and the infertile time a woman has and... every Catholic's favourite topic; HOLY SEX and every man's least favourite topic PERIODS!
Sorry, there is no real nice way to describe this, I’m going to be gory, so the weak should turn away at this point.

Let us think about the Jews for a moment.
(Ah thanks for the bagels and the fish and chips Jews!)
We need to look at Leviticus.

BLOOD, DISCHARGE and SKIN DISEASE
Many of the "unclean" suggestions came about because it simply protected the healthy. Those with leprosy were isolated from their community to keep the healthy clean from the disease.
Also... because they didn’t have washing machines and household cleaning items, houses and clothing had to be somewhat destroyed if containing the disease.
This is not much different today. We have isolation rooms in hospitals for extremely contagious diseases and infections for which we don’t have cures. Often anything put in contact with the person is also discarded. Think of precautions against HIV as an easy example.

NORMAL DISCHARGE
Leviticus does not deny that women's bloody discharge or a period is normal. In fact it even uses the word regular.
Regardless of this, you don’t see women exactly handing around their soiled sanitary items, sharing their soiled tampons etc. this would be unhygienic. Women make no aims to bleed all over places etc. we have these products for a reason... to contain and discard.
If it were clean there would be no problem sharing/reusing these products etc. but I’m pretty sure it is not clean.

BIRTH and DISCHARGE
The discharge a woman has after birth is (as far as I know) the same as a really long, really intense period. So same issues just for a longer time.

BLOOD, BIRTH, SIN and DEATH
During the fall (gah thanks for nothing Satan, Adam and Eve) we received pain and death. It comes directly from Genesis that we learn that painful birth, death, and illness is caused by original sin. God did not create an unclean world. It was sin that created this conundrum. So... whenever having a painful period... blame sin!

HYGIENE
It makes sense hygienically why one would not want to be in contact with blood and discharge from bodies. It can contain diseases such as leprosy, HIV, STI's, infections.
So what of the sacrifices and rituals?

WATER CLEANSING
Water is pretty great. We wash ourselves in it to be clean. Indeed, when a woman is menstruating she should be washing more and definitely should be trying to keep her clothing and bedding free from stains and clean. To just bleed everywhere would be horrible for everybody and to not clean it up would be disgusting.
We wash with water to be clean from disease, smells, and protect from germs/things unclean.
We wash with water ritually in religion to be cleansed from sin and protected from sin. If we think of sin like we think of germs we start to get a better picture. There is the germs that we already have and the germs we are at risk of getting. We wash to get rid of germs and potential germs. We wash to get rid of sin and to protect us from potential sin.

OUR LADY
Our lady went to be purified because it was humble of her to do so. Not only that, it acknowledges and emphasizes for us that temptation is real. Mary was indeed free from sin but she fought Satan. Satan would have really tried to tempt Mary who remained untempted because she followed the rituals God asked of her to do. Mary saw the importance of this ritual when it came to bestowing on women the graces to avoid sin. Mary didn’t need to be cleansed of sin, but she knew she needed God and grace to defend herself against sin. One could potentially say that the sacrifice she offered, the turtle doves and her son's foreskin (let's not forget the first shedding of blood from our LORD) is preparation for the sacrifice she would make looking on to her son's crucifixion.
She is full of grace because she knows she cannot do this without God, sacrifice is necessary.

DISCHARGE and INFERTILITY
There is a period during and after menstruation as well as during and after birth when a woman has this discharge that is hostile to producing life. It's quite amazing how the hormones work in a woman to prevent her from wanting physical contact at this time. I stress it is not impossible for a woman to get pregnant shortly after birth or when menstruating but it is scientifically improbable.
Married men and women are called to have holy sex which is open to life. It would not be open to life to have intimate relations during this time of discharge.
Essentially a woman's body is not meeting its full potential to carry life when she experiences discharge post birth or in menstruation.
This is why a man who makes contact with a woman's menstruation also becomes unclean.

SEMEN TOO!
Ah feminists do not get angry! Women aren’t the only ones whose discharge is unclean. Men's semen is considered to be unclean too and for similar reasons. Semen upon evacuating the body is unclean and the man from whom it evacuates is considered unclean ritually too. Think about it... semen can carry disease, and if not placed in the female's uterus or preserved artificially is generally infertile. If a bloke messes his sheets/clothes with his semen he should also clean himself and his sheets/clothes, it's basic hygiene. But also, semen loses its full potential to create life once out of the body and if not in another, so women in external contact with semen are unclean too.

SACRIFICE
So we understand water what of burnt sacrifice?
Ah fire purifies too! We think about boiling a kettle, killing the germs that can be found in water. We need fire to do this, we need the heat. It's also a method for discarding waste.
Leper’s would burn their soiled clothes and discard infected objects from their houses in order to be rid of their disease.
Ah but ritual burnt sacrifice can be understood when understanding the rituals passed to Moses.
The burning would be to cleanse from sin, as an offering of atonement, and as thanksgiving.

SACRIFICE BIRTH AND CATHOLICISM
The Jewish sacrifice was the lamb or the doves or the animals burned.
The Catholic sacrifice is Christ on the cross and in the blessed sacrament.
We do not need to make this burnt sacrifice anymore because of Christ but the reasons for sacrifice remain which is where the churching of women comes into play.
Birth is a risk, it can kill a woman. We thank God for our renewed fertility when the infertile period after birth is over and we thank God for our continued life post giving birth. We offer up the pains of birth in atonement for sin (not necessarily our own but sin in general) and we are finally clean and ready to carry about daily life without bleeding everywhere.

DISCHARGE, SEMEN and MENSTRUATION
This is where my question remains.
What of discharge, semen and menstruation outside of birth today? Does the modern ceremony exist for this nowadays and why, why not? Could it be because of confession, anointing of the sick, modern medicine, and because Christ died for us?
Once upon a time a man who had secreted semen and a woman who had her menses could not be in the sanctuary but both these people could have the sacraments these days. Though... if the man secreted semen due to mortal sin he would need to go to confession first.

In short we don’t believe that marital sex that is open to life is sinful and the reason behind this ceremony. (Augustine might have said otherwise at some stage.) Hopefully my points above help to make sense of this ritual though.

@charlesw
Be that the case then should we terminate the use of holy water and incense and the Easter vigil fire? All things still done in modern churches today.

As we are occasionally reminded on this forum, it is often better to respond to the original point rather than express all the reasons why it shouldn't be done, or what's wrong with it in the first place.

In answer to the OP, Stimson, I'm sorry - I don't have any resources for the ceremony of blessing women following birth. (Rather late now, anyway, as you wanted to use it last week!) In my experience, that is typically more of a private ritual and I've not encountered music for it in the past.

From the discussion regarding implications of the ceremony, the concept of ritual uncleanness was never my own take-away from the blessing, nor my understanding from the mothers who in my experience have received it.

Personally, I see this type of thought process as a manifestation in our society of the tendency to be offended by everything conceivable - usually while completely twisting the point around. In this context, that women might wear a veil while men do not; that women might wear skirts while men might wear trousers; that women might receive a special blessing that men do not - becomes an opportunity to bash "the Church" for being out of touch with the times and somehow denigrating the role of women. In reality, the Church recognizes our Lady as the first of all saints; encourages a high regard and respect of women; and recognizes them to be the heart of the family.

That Protestants might have contempt for the Church and its corresponding rites is one thing... to hear it expressed by Catholics, themselves, is truly unfortunate and not a little disheartening.

Our blessed Mother and our Savior are legitimately the only two with no need for purification from sin... (of which ritual uncleanness is a figure) and yet both took part at various times in ceremonies of purification without argument or contention. Perhaps we might benefit from their example without trying to justify ourselves by claiming the times have changed.