I would agree with some of the comments made earlier about following the Moraghan lead as I have been trying to sort out my own family lineage with similar problems to yours. My family is Moore but going back to my g. g. grandfather I have recently found out that his family name was actually Moraghan and not Moore. On the UK GRO records out of 9 children, 1 was registered as Morican, 6 as Moore, 1 as Morighan and 1 as More. On the Roman Catholic records for St Peters, Broad Street, Birmingham, 3 are baptised as a Moraghan, 3 as Moore, 1 as Moriken and 1 as Moor. My g. g. grandfather, James (Moore) Moraghan married Ann McLoughlin in Birmingham in 1850. James’ father is given as Owen, farmer, Roscommon. Next door to James and Ann in Birmingham on the 1851 census was Roger Morgan and his wife Catherine née Burke. However, on further research I have found out that Roger Morgan is actually Roger Moraghan and he is also from Roscommon (likely to be an uncle of James – research still ongoing) and his wife Catherine is from Galway.
If your Thomas is the one mentioned above then it could well be that his parents are the ones who died in 1901 and 1908 in Birmingham (Bridget Moraghan 1901 at 73 years old, volume: 6d Page: 126 and Thomas Moroghan 1908 at 80 years old, volume: 6d Page: 50).