Depends on whether you're keeping the same layout or need plumbing and electrics moving/redoing. My strip and start over bathroom was £8k but they redid the layout and I had loads of tiles and some fancy fittings, and they took care of everything (badly, it took 8 weeks, but I didn't do a thing).
If you're willing to strip out the old bathroom & take the detritus to the tip, it can save costs, as can buying the stuff yourself and just getting someone in to fit. You should be able to get the fixtures (bath, loo, sink, mirrored cabinet, basin tap, bath mixer, shower, lights) and decor (paint, flooring, tiles) for £1,500 or less if you really shop around and buy cleverly, waiting for sales and not using just one shop – the key thing to spend money on is the fitting, & things that might break. I.e. spend your budget on a decent shower, save on the tiles.
IKEA does nice posh-looking vanity units with big sinks, and other bathroom bits (I probably wouldn't buy a shower from them though); shower curtains and rods are cheaper (and nicer, IMO) than screens, etc.