Good luck then, brookeratheber, I think you will have your work cut out though unless the Aunt can remember anything more. I think you need to trace William's movements before active service before doing anything else. Even then, as Freebmd records give only the mother's maiden name, not her first name, you will still have lots of records to eliminate. As people have said, she could have been married or widowed, in which case the baby would be registered in her married name and her maiden name would be given - so really you would have to consider all babies born in that time span. You also need to remember that if the baby was adopted he/she will have had their name changed after that.
I think a better way would to have letters published in papers where he was, saying that you are looking for anyone who knew William Baker, hopefully that might lead to something.
I don't mean to sound negative, just trying to be logical.