Are you intending to move the existing boiler, or replace it and move the plumbing? Guessing from what you've been quoted it includes a new boiler. The job could range from very easy to very difficult/impossible, depending on the property and the existing heating system. Also, there's a significant range of cost in the boilers themselves. But you largely get what you pay for.
4,000 doesn't seem totally outlandish (around 2,000 for the boiler and bits, and as much again for labour), but I wouldn't touch British Gas for that sort of thing. They recently unilaterally announced they were no longer putting their fitters through the CORGI scheme*, because their training was so good it was unnecessary.
30 years ago we got married and moved into our first flat. I smelled gas, so called BG. They came out, twice. They condemned our perfectly fine gas fire in the dining room, but failed to spot the THREE leaks under the kitchen floor that I found later (and fixed). Complete cowboys. We've had a gas leak in our road for about eight years now. The junction has been dug up at least three times, and it's still leaking. The underground electricity sub-station is right next to the same junction. Methane is heavier than air. We're waiting...
I know several good plumbers in Bristol, but bear in mind the good ones are (a) more expensive, because they're worth it, and (b) usually very busy (because...). Also, this time of year is when everyone's heating breaks down, so they'll be extra busy.
If you want a couple of recommendations, drop me a PM.
E.
*It was around five years ago. I know it's not called CORGI any more, and that CORGI is/was a trade body anyway, not a standards body, but it had a small value. BG's decision was hubris, IMHO.