Have you thought about contacting the Salvation Army?I hear that they offer a service helping people to contact long-lost relaitives.
Also, have you got a copy of your birth certificate as I thought that the mother's (and often the father's name) were listed on that. You need to get a FULL CERTIFICATE - I don't think the dewtails are on the short version. Try www.ons.gov.uk (I think)
If not, go to www.google.com and enter UK Birth Certificates or something like that in the search engine.
You could also check all the birth entries for 1950 (most family history libraries have copies)for Appleby's born in that area. If you run into problems, contact me and I'll see if I can help.