Well, the actual fees to various bodies will be well under £1000.
The actual cost - well, if it were fairly well sorted already, then I could probably do everything in about 2-3 months. That's full time, which you won't get out of anybody, plus there will be inevitable delays for a variety of reasons. I am chartered aeronautical engineer who is also a Test Pilot; there are a few of us in the UK, but all are pretty busy. An extra cost is that you may need a fair number of spare bits of aeroplane to break in ground testing.
Say a few tens of thousands, spread over 6-9 months if everything goes well (and you get the right person/people on the job). 2-3 times that if it doesn't go perfectly.
Incidentally, I'm not going to recommend any individual aeroplane, but would strongly recommend buying the current "World Directory of Leisure Aviation", which is about £6, covers most of what's out there, with prices and performance. Also look on the BMAA and LAA websites for details of what is approved.
G