RevAMG -

The Psalter of the Ordinariate is the Coverdale Psalter.
None other is (or should be) used.

At Walsingham (the cathedral) we have since before the erection of the parish in 1984 sung the Psalm in directum to Anglican chant.
The entire congregation sing it, and sing it well.
Too, the chants vary from week to week, depending on the psalm - but this is nothing to fret about with our people.
This has never varied.

I really do not know whether this practice is followed throughout the Ordinariate or not.
I should think that it would be, but really don't know.
If there are any who sing the Psalm responsorially, still, they would use the Coverdale.

We never sing the Gradual Responsory from the Graduale Romanum, only the lectionary psalm, Coverdale translation, to Anglican chant.

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As an aside, Anglo-Catholics of old frowned quite royally on Anglican chant and spurned it as 'protty', and would have naught but the Gregorian psalm tones.
While I love the Gregorian tones I never shared this rather snotty and eccentric point of view.
Anglican chant is beautiful and is one of the jewels of our patrimony.
I should think and hope that all our parishes would sing their psalmody to it.

(Actually, one can sing alternate verses of a psalm to a Gregorian tone and an Anglican chant, fa-burden style. Care, of course, would be taken that they are tonally compatible.)

Thank you, Father, for your enlightenment.
Alas! It would appear that even the Ordinariate is not exempt from the old 'this or another suitable.....' fig leaf.

Noel, in Masses according to The Roman Missal and according to Divine Worship, the congregational response may be omitted. The rubric for this is found in the General Introduction to the Lectionary at no. 20 (and cited in no. 22 of Divine Worship's Rubrical Directory):

20. As a rule the responsorial psalm should be sung. There are two established ways of singing the psalm after the first reading: responsorially and directly. In responsorial singing, which, as far as possible, is to be given preference, the psalmist, or cantor of the psalm, sings the psalm verse and the whole congregation joins in by singing the response. In direct singing of the psalm there is no intervening response by the community; either the psalmist, or cantor of the psalm, sings the psalm alone as the community listens or else all sing it together.

I think in Melbourne we used the lectionary and it wasn't sung when I was there because it was just starting and we had a speech expert who read it with amazing elocution but I'll ask current directors over here if you like.

tsoapm, the current General Introduction to the Lectionary (GIL) was published in 1981. In the [brief] history of the Ordinary Form of the Mass, responsorial singing has been the norm for the Ordinary Form. I don't want to get into the Extraordinary Form and the much longer history of the Gradual, but for the Ordinary Form, the GIL presumes the norm of responsorial singing (notwithstanding the plethora of options available, discussed elsewhere on this forum).

Divine Worship: The Missal has been in use only since Advent 2015. The normative Anglican custom of singing in directum is preserved but the Rubrical Directory also permits other options as outlined in no. 22 above. In fact, there is a footnote to no. 22 of the Rubrical Directory that cites GIL, no. 20. Hence, the relationship between Divine Worship and the Roman Missal (and its rubrics) is interesting. For example, the Rubrical Directory states:

7. The liturgical norms and principles of the General Instruction of the Roman Missal are normative for this expression of the Roman Rite, except where otherwise stipulated in this Directory and in the particular rubrics of Divine Worship. This present Directory is intended to provide instructions for those areas in which Divine Worship diverges from the Roman Missal.

This is why the GIRM is printed in full at the beginning of Divine Worship. Furthermore, the Rubrical Directory gives some leeway as to what can be sung for the propers of the Mass (cp. GIRM, 48, 61–62)

14. The given texts of the chants may be replaced by the chants of the Graduale Romanum or by musical settings of the Graduale which rely on a different translation of the same texts. The Gradual and Alleluia may be replaced by the Responsorial Psalm and Alleluia of the Lectionary. In addition to, or in place of, the Introit, Offertory, and Communion, an appropriate hymn may also be sung. Likewise, the given text of a Sequence may be replaced by a prose or musical setting which relies on a different translation of the same Sequence.

In the end, Divine Worship gives options (and does not normally stress one-thing-over-another) so as to respect a divergence in liturgical experience among Ordinariate communities. One community's liturgical experience may be closer to the Anglican Missal tradition whereas another community's experience may be closer to the Roman Missal as revised following the Second Vatican Council. Hence, Divine Worship allows for a variety of expressions that may-or-may-not follow the other (earlier) rubrics precisely.

Father AMG -
Are you an Ordinariate priest?
A friend of the Ordinariate?
Other?

Actually, one can sing alternate verses of a psalm to a Gregorian tone and an Anglican chant, fa-burden style. Care, of course, would be taken that they are tonally compatible.)

Yes, we do this with doxologies at our Mass in the EF.

At the ordinariate in my city, they use the psalm tones, but it's not what is preferred by the powers that be. I won't say anymore on the subject.