My grandmother, or as I knew her Nana (b. 1919), was most likely illegitimate. While her parents lived together, we believe they were never actually married, as he had already had a wife and had never divorced her. Either that or he was a bigamist. Growing up working class in London, just how shameful was something like that.
Can you give a useful answer to the question of how shameful illegitimacy is in your nearest large city now?
The answer to your question is going to depend on their local community, and maybe on the attitudes of their extended family. Items of your family oral or written history are going to be a better guide to their experience than more general historical sources.
The restricted access to abortions and divorce did mean that many families, whatever their professed attitudes to marriage, had illegitimate members. My quick look suggests an illegitimacy rate of about 5% across London as a whole, but I am not sure a second family would show as illegitimate -- the children might well be registered as the legitimate children of John Smith of 15 Green Street secure in the knowledge that no-one will connect him with John Henry Smith of 20 Crown Street a hundred miles away.