Thank you rlw254 and brigidmac for your advice and suggestions. Unfortunately, my Grandmother had only one son - my uncle - and he died several decades ago now. His son is also now deceased.
My Grandmother was born in South Shields, which was then part of Durham, in 1895, and as far as I am aware no affiliation orders would now exist. However, if I am wrong then I would like to know about this as I would definitely explore such as avenue if it was possible.
You are fortunate, brigidmac that a court record exists and so you have been able to get closure about who your great Grandfather was.
I have tried all the Church of England Churches in South Shields without avail when looking for a baptism. I would think it would surely have been considered quite important at that time for a baby to baptised and if only I could find out where - perhaps her father might be named.
This was lucky that you did not miss out on finding your own Grandma's baptism. A similar scenario happened to me when I was looking for my Grandmother's school admission record. I had looked at a particular book and found nothing. I closed the book and was just about to hand this back to the archive staff when a strong thought suddenly occurred to me that I needed to look again. On the second search I found my Grandmother's name, her exact date of birth and the name of her Grandfather under the parent or guardian column. Too many coincidences for this to have been anyone else but my Grandmother. She was age 12 at this time. It was also recorded that her previous school had been in a different named area and I know that this area was where her birth mother was living. One intriguing thing was that the address recorded at the South Shields school was not the address of her maternal grandparents and family. I did detective work on who lived there at that time and found that a widow and her family were long time residents at the address given. I did wonder if one of the widow's sons may have been responsible for fathering my Grandmother and that she may have housed her as she was a possible Grandmother - but on checking one son was too young and the other died before Grandmother would have been conceived. So I suspect Grandmother may have just boarded with the widow and family as a lodger perhaps. It is all such a mystery! I do hope somehow, someway, I may be able to get closure on this brickwall mystery eventually.
Thank you both again - I really appreciate that you have both made an effort to try to help by offering possible helpful and advice and suggestions. This was very kind.