I have a part p electrician who I've been told about.
Well - I hope he is actually recommended by someone you trust. I agree, get other quotes to make sure you aren't being taken for a ride, but personal recommendations are always the best way to find a reputable tradesman. If you're having to go ahead without much in the way of those, or references, don't put any store by registration itself - sadly it is possible to become registered with woefully inadequate qualifications and zero practical experience. You don't have to spend long here to see people cropping up who are registered and "qualified", but who are clearly seriously incompetent in reality and who should not be charging for their services.
You are looking for someone to rewire a house, and it may surprise and dismay you to learn that it is quite possible to become a "competent person" without ever having done that before, and without having acquired any of the practical skills needed to do it without half-destroying your house in the process.
It's your money, £'000s of it, and you have every right to ask prospective tradesmen what their qualifications and experience are.
Just being listed here is not a good enough guide. No genuinely experienced electrician, with the "full set" of C&G qualifications will mind you asking - in fact he will wish that everyone was like you.
I feel sorry for people who have been misled by training organisations and (shamefully) the Competent Person scheme organisers into thinking that a 5-day training course, a couple of trivial examples of their work and some basic understanding of how to use test equipment will make them an electrician, but not sorry enough to agree with them trying to sell their services to Joe Public.