When you say he was born and raised in the 1930s I'm not clear about when he served in the Army. Presumably this was during National Service in the 1950s? When you say he finished school in his early 20s I assume this means that he went to university. If that is so he didn't have much choice about the National Service bit. As for going into the Army and getting commissioned, perhaps he went to a school with an ACF or CCF, and so he had some early military experience which made him more suitable for the Army rather than, say, the RAF. Then a good (university?) education plus some cadet experience would have marked him out as possible officer material. Since the Irish Nationalists such as the IRA were not particularly active during the 1950s his political background is unlikely to have been an issue, assuming the Army even knew about it.
I doubt if it would have caused any real problems with his family. They can't have been that anti-British if they were living and working in Birkenhead. He was most likely to have found himself deployed to Malaya, Borneo, Aden or West Germany where his political leanings wouldn't have posed a dilemma for him. It would have been a very different situation had he been serving after 1969.