Several things come to mind here and I hope none of it sounds too simplistic.
1. Have you considered that they may have gone out of the area to have the child - we had one who I knew was born in a certain year and even the quarter they were born in, but they had been sent to Blackpool to give birth!! That took a while to find I can tell you as we have no links to Blackpool!
2. If you have a year of birth and a possible location, have you tried searching for the person just by their Christian name(s) and then looking at the surnames, to see if the child was registered in the fathers name? If you don't know the fathers name, see if any names ring any bells.
3. If that fails, again try just with Christian name and search through each quarter again to see if any name jumps out at you. It's always possible someone was 'economical' with stating the year of birth so widening the year search might help too.
1912 should have the birth showing the mums maiden name, but sometimes in searching , less is more!
In sensitive cases, you may not wish to post the name on here and we all understand that, but if you wanted too and they are definitely deceased now, then I'm sure people would be happy to help search with you. Best Wishes