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The tune is called "Cobb" and can be found in the Hymnal 1940. The number is 134.

We sing this tune to the St. Stephen's Day hymn for example. I could not make out their words, but perhaps it was "Come, pure hearts, in sweetest measure" translated from Latin by Robert Campbell (another convert from Anglicanism).

As Steve noted above we also sing the tune "Evangelists" (which also is known by a German name... perhaps "Ach, Herr") to these texts from time to time.

I think they are singing a translation of Lauda, Syon. In one of the translations we would use two hymn tunes for the translation because of the change in metre. The video ended before I could hear the second tune, but with one note I am trying to remember.

I wonder how many are left who still wear the St. Stephen's House cassock. Some anglo-catholic Anglican priests from Nashotah House and elsewhere also wore the cassock, and thereby identifying their allegiance to the Catholic wing of the Church of England. The St. Stephen's House cassock was beautifully tailored, and I treasured mine.

I'm also remembering another text of the same metre. Adam of St. Victor 'Plausu Chorus Laetabundo' I don't recall all the words, but it concluded with beautiful words in translation, something like:

Four the wheels which Godhead carry
On these staves the Ark may tarry
Four flow Eden's streams again.
Happy streams, God's gift renewing,
Man with Sacraments bedewing
These the nations' life sustain.

Firmly is God's temple grounded
On this four-square basement founded,
Ruin ne'er can it betide
May He in this House delighting
Dwell with man in bliss uniting
God and man for aye allied.

To be certain would need to find the hymnary from which it cometh. I suspect 'Songs of Syon' was the book. I suppose today the last line would be rendered "God with us for e'er allied" which of course would be wrong since it is not speaking about us alone but all humanity.