What is the basic meaning of 'dozy pillock'?

Solution 1:

In the context you supplied, dozy means 'stupid'.

Pillock has its origin as one of the numerous slang terms for 'penis', but today I think that association has receded from most people's consciousness, and it is now a derogatory term that usually conveys the meaning of 'a person who is an idiot, or who has done something stupid or silly'.

Solution 2:

Having been born in the UK, I and many others have used the term Dozy Pillock for someone who has done something stupid or downright dangerous. Not knowing the full meaning or derivation does in no way deter people from making such a comment. It is akin to Dill or Dickhead or Bampot from Scotland.