I saw you posting of four years ago in the archives of genealogy.com.Maybe you have got further in your investigation already, but if not, then maybe I can help.
I think the first thing you need to do is to trace your Dorrell grandparents.For that, you will need to know their names, if you do not know them already.I found your father in the births index - it confused me at first because he was not registered until July 1955.However, the reference is 1955 Q3 (i.e. Jul-Aug-Sep), district Surrey Mid E., volume 5g, page 271.Your grandmother's maiden name was CAIRNS.That's all the information in the index - to go any futher you will have to send off to the GRO for his birth certificate.Try http://www.gro.gov.uk/gro/content/certificate/index.asphttp://www.gro.gov.uk/gro/content/certificate/index.asp. Certificates will cost you £7 each.The birth certificate will tell you the full names of your grandparents, as well as lots of other interesting stuff.
I hope you manage to make some progress in you project. If you can get back to 1901 then it suddenly becomes easier because all the census returns are available on line.
Write to me "Kevin dot Dorrell at pt dot lu" and let me know how you get on.Good luck, and who knows, we might be related!