What's the name of this setting for the Angelus? Here's a YouTube video of an English-language setting of the Angelus that appears to be popular among Anglo-Catholics (it's used at Pusey House)...I've not been able to find the name of the hymn tune.
Link: The Angelus
And here's the specific wording of this variant:

The angel  of the Lord brought tidings to Mary: And she conceived by
the Holy Ghost.
Hail Mary, full of grace, the Lord is with thee;
blessed art thou among women, and blessed is the fruit of thy womb
Jesus. Holy Mary, Mother of God, pray for us sinners, now, and at the
hour of our death. Amen.
Behold the handmaid of the Lord: Be it unto me according to thy word.
Hail Mary, &amp;c.
And the Word was made flesh: And dwelt among us.
Hail Mary, &amp;c.
Pray for us, holy Mother of God. That we may be made worthy
of the promises of Christ. Pour forth, we beseech thee, O Lord, thy
grace into our hearts; that as we have known the incarnation of thy
Son Jesus Christ by the message of an angel, so by his cross and
passion we may be brought to the glory of his resurrection. Through
Jesus Christ our Lord. Amen.
It may not have a name. The melodies of the tune are rather standard chant formulae used in Anglo Catholic circles, but not necessarily specifically for the Angelus, so it may not be a formal, named hymn tune, or if it is a named tune, it may be in very limited distribution. You might try to contact the organist of the congregtion mentioned in the comments on the video, if all else fails.