Try Riga Interiors - Vitaliy is the owner. Tell him Boon sent you!
Used them for a 2 bedroom flat years ago, and also a 5 bedroom house recently. Full renovation. They can do all the jobs you've listed. I've been very happy with their work, and the price was reasonable too. You'll have to source and buy all the materials though (tiles, bathroom fittings, bath, shower head, etc) as they're not "full service" in that sense.
Carpet though, I would consider just using the carpet company to install. Probably cheaper, and better experts. I used Tapi down in Tooting, got an excellent service and good fit.