How many bathrooms do you have? I'm in the process of having a bathroom fitted and went the self organisation route but it's taken weeks and still not finished. If we didn't have other bathrooms would not have been doable to have this one out of action for ages.
The things that have taken longest are:
Plumbers merchants taking orders for stuff and then taking ages longer than the lead time to deliver, e.g.
'In stock' shower not turning up for three weeks with periodic this has not been despatched emails
Duravit suite which I checked and rechecked the lead time for and as soon as they had taken my payment again turned out to be three week lead time though to be fair they at least gave me a delivery date and stuck to it.
So if you're in a hurry I would try to select in stock and physically on the premises stuff if possible.
I have a bathroom fitter who does the plumbing and tiling. He has charged me £3k which included ripping everything out, 11m2 of tiling, plastering the rest, putting in false wall, refitting in the same spaces. £6k sounds a lot to me but I'm not in SE. My £3k also covered him subcontracting the electric work, but the wiring for fan, mirror etc was already there.
It took me a week delay to get my own decorator in. Bathroom largely tiled so cost was £100, but I was lucky he could fit it in between other jobs. Think about who will do this as I wasn't keen to do myself round new tiles etc.
I'm currently waiting for the floor fitter which I've organised separately. This is for amtico and therefore a specialist fitter. I'd suggest you think about flooring now. If it's not tiles your fitter can do get something chosen and a specialist fitter provisionally booked. Floor ideally needs to be done before the loo and basin are fitted.
The most time consuming thing for me has been a vanity unit. I'm having it made by a local bespoke furniture place as the space is awkward. They couldn't measure precisely until the bath was in and then their lead time is 5-6 weeks. I also had to order the granite for the top separately and suspect that eventually uniting the granite and the cabinet will drag things out even further. I guess my advice would be don't do this unless you also have a very awkward space.
If you haven't chosen tiles yet order as many samples as possible now. After searching on here for recommendations I ordered from Tons of Tiles (rumoured to be same tiles as Fired Earth, look at their crackle glaze), Walls and Floors and Mandarin Stone. I didn't end up with what I thought I liked best on ordering. Force yourself to get a load of samples - negligible cost - and get tiles picked out and ordered. Hopefully your fitter has told you how many you need and you have somewhere to store them.
I spent ages looking at mirrors; partially because of awkward dimensions. The light fitting also took a while to select. Lights are from castlegate lights. They have a vast amount of choice. In my experience light fittings are a really hard thing to choose so again invest some time so you have it ready.
Also think about any small things you want. You definitely need a loo roll holder but do you want anything like soap dish, toothbrush holder etc fitted. If so again get them ordered.
Will you be having a radiator as well as underfloor heating? Again I spent ages looking at these as I wanted something that looked nice rather than functional but adequate temperature was also vital and I'd read a lot about people choosing things that weren't warm enough. I actually bought my radiator off eBay in end from a guy who had apparently just closed down his bathroom shop and was getting rid of a last few bits. I'd picked what I wanted elsewhere and then just had a quick search and it came up. Cost £150 instead of the retail of £400 ish and as he physically had it came quickly. You may not have time for this but a quick Google can save a bit of cash.
On suite and fittings my theory was the labour costs the same whatever the quality and as the experience is quite painful to go through it is definitely worth paying for decent quality makes. It would definitely be possible to source things for less than I did, but I'm hoping things will last longer and also that I've put stuff in that fits with the overall value of the house if we did come to sell.