My grandmother was born in a Sussex workhouse in 1902 with "no father" and her mother went to great lengths to conceal the illegitimate birth - Nanny didn't discover until she reached retirement age that she had no birth certificate (of course she did, but this was still kept from her and caused no end of problems claiming a pension).
I now have a couple of possible candidates for the father but, short of putting a drawing pin on a chair or snagging a hair from someone of the same surname and subjecting the results to genetic tests, I suspect I won't get any further. AND, of course, these candidates could be completely wrong!
In my case, although I would be interested to find the blood line, I am tempted to say "biology be damned". Another, very lovely, man was her de facto father for 60+ years (and my mother's grandfather for 24 years).
Having said that, you may find documentary or circumstantial clues to the father. I shall continue to look for my g grandfather, but won't make it a priority.
Deborah