If your son was born from an illicit affair he would not have the same Y-chromosome as you and you would both only match on a few markers and might even be in a different haplogroup altogether. Family Tree DNA now has almost 200,000 Y-DNA results in its database. Both of you might still get matches within the database at 37 markers. However, even if the surname of the real father was not known, if he has close matches at high resolution with other surnames this would provide clues about the potential surname of the biological father. You would still have to do the paper research to work out the actual connection.
Some adoptees have been able to trace their birth fathers through DNA testing. Dick Hill's website has some useful information:
http://www.dna-testing-adviser.comThe success rate is currently about 30% but I think it would be lower for people in the UK.
There have even been cases of sperm donors finding the identity of their fathers through DNA testing:
http://news.bbc.co.uk/1/hi/health/4400778.stmWith illegitimacy, adoption, etc, it is all a matter of luck as to who else is in the database and whether or not they have the information which might help.
Most people however use DNA testing within structured surname projects which are aiming to test all lines and work out which lines are genetically related.