At present someone that does DIY work cannot get it checked easily, so often just hide the fact that the work has been done. Or designs the work in a way so as to avoid Part P, e.g. adding lots of sockets to a current ring rather than putting in a new ring.
I would expect someone like “DIY Dave” to keep doing what they have always been doing and not give a whathaveyou, so not get anyone in to check the work regardless of the regs.
However I think the much bigger issue are people being charged for sub-standard work by kitchen fitters etc and believing they have had a good job done. Personally I would like to all work that have been charged for (directly or as part of another job) having to be notified.