There is no set hymnal. It is up to each parish. Most select hymns from Anglican hymnals, especially those pre-1960. St. Anthanasius, the Anglican Use parish in Boston, uses the 1940 Episcopal Hymnal supplemented as needed to fit the gospel reading of the day. Many of the Wesley hymns and other older Anglican and Protestant hymns are universal and quite compatible with Catholic teaching.
That being said, you might not actually see any hymnals in pews. Bulletins in many Anglican churches differ from Catholic bulletins and often include along with announcements often an order of service with all the readings, prayers, responses, and hymns for the day. This allows the music director and/or pastor to take the hymns from whatever source he wants. It also means that you don’t need thick, confusing (often cheap looking) missalettes and it gets the bulletins and its announcements into everyone’s hands.
One last point, since most of these hymns are generations-old (and survived the test of time) they are well beyond copyright, if they were ever copyrighted at all. So, there is no problem freely reproducing them.
