On the assumption your Dad's birth was first registered under a different surname, I would suggest you collect for a period of say 3months near his known birth date, all instances of the Christian name of Derek from the birth index. I know this will be an arduous task but the name Derek is not so common and hopefully Derek was name he was registered with. Also Mum may have moved away to have the baby, so all places within reason.
(If you knew Roger's approx birth, you could search until you found a corresponding surname. Very arduous, I know.)
Keep this list of surnames when doing further research. I broke down a brick wall by collecting possible surnames in the 1860s, but unlike you I had the census indexes to then search.
Do you have a possible year that Mum and Step dad married? Another arduous task, if you are desperate. But a corresponding surname may crop up.
The very best of luck, Gazania