My lord I don't know where to start . My grandad had a full on affair during the Second World War he was based in Beer. He got his lady pregnant . The war ended he returned to Coventry . Asked his wife for a divorce. She refused . They went on to have my dad. Their is about 4 years between the half siblings. My dad got this info from his late grandma. What's our chances of trying to research her. ??
Perhaps you could write to the letters page in the local paper, and give some general details. Someone might remember something if prompted, and give you a starting point.
The Salvation Army provide a service for tracing long-lost relatives and it could be possible that she is searching too.
There's a website called FreeBMD where you can search on births in a particular area and timespan - if you find out which registration district births in that locality would have been registerd at, it might help in your search. Your grandad may or may not have been registered as the father of the baby, and the baby may or may not have his surname. Sometimes babies born out of wedlock were given middle names that can offer a clue, especially if there's an uncommon first name or surname used by your side of the family.
Look on GenesReunited - there could be someone already researching your grandad's name.
You could also try the family history society for that part of the country. Good luck!