ICEL's permission policy is here:
http://www.icelweb.org/PubPolicy.PDF

and the last page has their policy for internet use; it's fairly broad, so my amateur opinion is that you don't have to ask for a license, provided that you give the necessary acknowledgement.

CMAA's "The Parish Book of Psalms" has a cappella settings that can be used at no cost under a Creative Commons license. See https://musicasacra.com/pbp for info and downloadable PDFs, and the hardcover version.

A list at the CMAA site describes projects in which composers have written settings for psalms, gospel acclamations, etc. Some are licensed under Creative Commons rules, so permission is usually automatic, as long as you provide the appropriate acknowledgement.
https://musicasacra.com/music/english-propers/

I would say the ICEL license as stated on p 25 “Use of ICEL Materials” doesn't allow recording or broadcast. It explicitly prohibits “other form”, and clearly the licensed form is only textual. So you can make copies for distribution or rehearsal, but not recordings or broadcast. I think if they intended to give common license to broadcast of their works they would have said so explicitly.

(Neither the British or US copyright law restricts the use of works in religious services as such: this is only about copying, recording, and broadcasting.)

I think ICEL ought to place all their liturgical products outside the reach of any copyright law, because they aren't in it for profit and the canon law already protects the liturgy from corruption and misuse.

But they don't.

For what it's worth, I called the St. Michael hymnal people about thus same question, do we have permission to use anything in the hymnal for live streaming etc? They said thereis no extra cost.
However we bought a church full of hymnals....

ServiamScores- one would think that after the parish has invested thousands of dollars in hymnals that one would be able to livestream the music in them in the parish's Masses, etc. It's nice to know the St. Michael Hymnal publisher is so generous. However, we have OCP hymnals, and although we can use whatever we want from them during non-livestreamed Masses, we were told we needed a livestreaming license to use any of the non-public domain music from their hymnal (including the altered versions of texts that should be in public domain but aren't) when we livestream. The licensing service is so expensive that after we read about Noel Jones' new hymnal, A Catholic Book of Hymns (which requires no permissions/licensing) we decided it was best to just buy copies of that and not use the OCP hymnal at all, hence my original post. With all the necessity for new ways to reach parishioners during a pandemic, I'm not sure why churches have not complained that livestreaming Masses should be covered under the same copyright exemptions as live performances in church.

To be fair, we are in an awkward time of transition. The model which has stood for a few hundred years now, had texts which were freely available but we paid for printing and binding, if that was what we wanted. Then, notably in the US constitution, the moral notion of copyright took hold, and human greed has gradually transformed it from a codifying of a moral entitlement into a tool of Mammon. Which has unfortunately ensnared the Church.
Now we have a new situation, where printing is available cheaply and ubiquitously, and neither the civil law nor our moral system has evolved to cope.
NB the system was not always simple or moral e.g. Tallis, Byrd and the monopoly of music printing.

1. Simplest solution is to not livestream.
2. Remember that texts are copyrighted separately from music, so even if you are doing all public domain music, the lectionary readings are also copyrighted and cannot be broadcast or recorded without permission.
3. If you do livestream, use only public domain music, which would necessitate only singing Mass parts in Latin.
4. You could also not livestream, but simply record and edit out all the copyrighted material before posting online--which would leave only the homily and any public domain music.
5. You can pay a lot of money for a license to do everything by the letter of the law, which also requires the requisite reporting, etc. A lot of work.
6. Do whatever you want and hope not to get sued. Chances are, the USCCB won't sue a parish for livestreaming a Mass, but there's always a chance.
7. Seriously reconsider option 1 and save a lot of money, time and headaches.

This is false. There are plenty of other things that are PD or permitted free usage.

Not False.

I was referring specifically to the words of the Mass. The Roman Missal as well as the Lectionary in English is copyrighted. While I'm not aware of the USCCB or ICEL suing anyone for livestreaming a Mass, they technically could if they wanted to. While there may be musical settings that are in the Creative Commons or Public Domain the texts themselves are not--though they should be.

I said use only public domain music (assuming you don't want to pay for a license). The point is the music and texts have to be considered separately. Yes there is plenty of good public domain (or creative commons) music that could be used. But we have to use approved translations of the Mass texts. One could not do their own translation or use a traditional translation such as "Lamb of God, who taketh away the sins of the world." For a concert that would be possible, but for Mass, one is bound to use only the approved translations in the Missal which are copyrighted.

Like I said, it's unlikely that the USCCB or ICEL would sue a parish for livestreaming Mass without first seeking permission to do so, but it is possible according to the letter of the law.

Consider that Lumen Christi was able to post "free scores" online using the Revised Grail translation of the psalms, because GIA had given them permission to do so. But now that they are using the Abbey Psalms which are owned by the USBBC they have to put them behind a paywall.

In my opinion, the words of the Missal, scripture translations and any other official liturgical texts should be placed under Creative Commons. That still protects the texts from unauthorized changes, etc. but it allows the Church to use the texts without having to pay for them. Consider how difficult it is for an individual to pray the Liturgy of the Hours. What if a parish actually wanted to have Sunday vespers prayed in common in the parish. There's no easy (or inexpensive) way to do this with the current copyrighted texts.

it's unlikely that the USCCB or ICEL would sue a parish for livestreaming Mass without first seeking permission to do so, but it is possible according to the letter of the law

Has anyone ever tried it out? Would be worth the case...

Lumen Christi [...] now that they are using the Abbey Psalms which are owned by the USBBC they have to put them behind a paywall

Not necessarily, they could publish the music for free and give it a footnote:
Psalm xxx, English translation (c) USCCB.

but for Mass, one is bound to use only the approved translations in the Missal which are copyrighted

Go ye into all the world, and preach the gospel to every creature - and have them pay you a copyright fee for subsequent use

The USCCB allows for the readings to be live-streamed, but they must be cited in the description.
While that Onelicense coverage will cover hymns from MOST hymnals, GIA, OCP and their ilk, it will not necessarily cover choral music, or organ music that is still under copyright.