A bathroom refit tends to produce quotes from three different sorts of business, and they are not quoting for the same thing. One is from a plumber. One is from a bathroom fitter. One is from the installation service that the merchant or the big retailer offered when you bought the suite. Working out which to hire depends on what you are hoping to walk into when it is finished.
A plumber's core work is the pipework and the sanitaryware: hot and cold supplies, waste runs, the toilet, basin, bath and shower connected and tested, the isolation valves in the right places. A plumber who does a lot of bathrooms will often tile and do the making good too, but it is worth asking rather than assuming, because tiling is a separate skill and some plumbers would rather hand it to a tiler they trust. If the job is a like-for-like swap in an existing room, a plumber alone may be all you need.
The one thing to watch is electrics. A new electric shower, a shaver socket, downlights or an extractor fan need someone who can certify the work under the wiring regulations, and a bathroom is a special location with its own rules. A plumber cannot sign that off unless they are also registered for it.
A bathroom fitter is a project trade. The good ones carry the plumbing, the tiling, the building work such as boxing in and moving a stud wall, and either do the electrics under a registration or bring an electrician in and manage it. You get one quote, one point of contact and one person responsible when the shower tray leaks a month later. The price reflects that: you are paying for coordination and for the fact that the fitter absorbs the risk of the trades not lining up.
Ask a fitter who exactly will be on site. Some are a single person who does everything over two weeks; some are a firm with a team and a project lead who visits. Both can be excellent. What you want to avoid is a fitter who subcontracts everything and is never on site, because then you have a project manager on a plumber's mark-up.
Installation through the retailer is convenient and often comes with a guarantee that covers both the goods and the fitting. The trade-off is that the installer is usually a subcontractor to the retailer, working to a fixed specification and a fixed time. Anything outside the spec, from moving a soil pipe to sorting out a rotten floor under the old bath, becomes an extra, and the person on site may not be free to negotiate it. If your room is straightforward and you want the guarantee, it works. If you suspect surprises behind the old tiles, you want someone who can price them on the day.
The honest test is how much you are changing. Same layout, new suite: a plumber, possibly with a tiler. New layout, walls coming out, a walk-in shower where the bath was: a bathroom fitter who owns the whole job. Standard room, standard suite, and you want a single warranty on the lot: the installer. Whichever route, get the first-fix plumbing inspected before the tiles go on, because that is the last cheap moment to fix anything.
Among the 3,554 plumbers listed here, plenty describe bathroom installation as a service and say whether they tile. Browse the firms in your area and read what each one says it covers, or describe the room and the changes and ask for quotes. Send the same description to all of them so the quotes can be compared like for like.