Have you checked on freebmd that I gave a link to for any females born in the quarter corresponding to your mothers date of birth?
You might need to try a few variations as some of her original details may have stayed the same for example her first and middle names as well as her date of birth. You could search without a surname just using first and middle names and perhaps the county where the family lived. They may have adopted her from elsewhere, but you have to start somewhere I suppose.

I would probably also be initially looking for the baby having the same surname as the mother, as it was likely that she was illegitimate. Then broaden the search if you find nothing likely.
If you would be willing to share her name and other details from her birth certificate (presuming everyone named is deceased) I'm sure there will be plenty of helpers willing to lend a hand. If you don't (which is understandable) then try freebmd first.
As Annie says, she could have been born in Scotland ... or Ireland, England or Wales.