I'm Irish, though not from NI, and it's my dad's name - always pronounced 'Ant-ony' though spelled 'Anthony'. In the UK, the only person I know who pronounces the 'th' is my SIL, and I suspect that in her case, it's not regional, but comes from a desperate desire to appear sufficiently middle-class - I think she thinks everyone else is mispronouncing it, including the parents of a young Anthony we both know. They will say 'Come here, Antony', and SIL will immediately say 'Come and have a sweet, ANTHONY', as if she's gently leading them the right way. 
Interesting question, though. Is there, as I suspect, a class as well as a regional element in UK pronunciations of the name? I know three UC Anthonys, from different parts of the country, and none of whom pronounce the 'th'...