Welcome to Rootschat. The original poster didn't mention a family surname so not sure if you are searching for the same people.
At the top of the Tyrone board are a few stickied threads for church records and other resources. Clicking on this link (
http://www.from-ireland.net/tyr/tyrindexrc.htm) under R.C. records shows that the following records for
Drumragh (Diocese of Clogher) exist and microfilmed copies will be in National Library (Dublin) and PRONI (Belfast)-
Baptisms: May 1846 – Nov 1846, Nov 13th, 1853 – Dec 22nd, 1880 - indexed
Marriages: June 1846 – August 1846, Nov 7th, 1853 – Dec 26th, 1880
Burials : May 1846 – Sept 1846, Nov 23rd, 1853 – Dec 11th, 1880
There's also a link to Sacred Heart Church, Omagh which might be the one you mentioned, although the congregation is certainly much older than 1899.
St. Columba's Parish Church, Drumagh (Church of Ireland) is listed on C. of I. website:
www.ireland.anglican.org/index.php?do=information.dioceses&page=showparish&id=11300If your family were in Ireland in 1911 their religion should be listed in the census:
www.census.nationalarchives.ie/search