Towel rail and UFH seems reasonable.
The fan and mirror there is a problem.
"Every installation shall be divided into circuits, as necessary, to: take account of danger that may arise from the failure of a single circuit such as a lighting circuit" that is just one of six, but it is the main one, the problem is lights use ceiling roses and other items only rated at 6 amp, so in the main we use a 6 amp MCB and RCD or RCBO on the supply, the MCB is faster than a fuse, so any fault will normally trip the MCB before it blows a fuse.
The use of lighting supplies for other things is done, common to find a TV aerial amplifier run from lights, and the fan, but it needs a risk assessment, if the item fails what will happen, so with emergency lights fitted in the house, if you lose lighting supply, there is no danger, so there may be a case where it is permitted.
However in this house there is very little light from street, and loss of a lighting circuit would present a danger, I would not supply any other item from the same MCB or RCD as the danger is clearly there, so this house all RCBO used. Last house street light shined in through the landing window so not so much of a problem, house before that, fitted emergency light above stairs.
However your main problem is to work out what the LABC inspector will permit, does not matter what I say, it is a notifiable job, so some one has to inspect and pass the work, even using plastic to fix wires can be seen as non compliant, yes some of it seems daft, used plastic raw plugs for years ever since the asbestos ones were banned, but your at the whim of the inspector, the equipment has to comply with the relevant standard, and is suitable for use in that zone according to the manufacturer's instructions for use and mounting. So unless the mirror states a bathroom mirror for use in x zone, there could be a problem. The same with UFH, in a bathroom only some types are permitted, some times one has to fit earthing mats over the heating cables. Also everything must be RCD protected.
To be frank, today I would not bother DIY in bathroom, too expensive to get LABC involved.