The potential problem is that with the door to the cupboard open the
horizontal distance from the edge of the bath to the boiler is about 40cm
and so it could be touched by someone standing (not sitting), which is why
I
wondered is putting a lock on the door makes the boiler, in effect, in a
separate room.

My understanding is that it does. The use of a "tool" (in this case a key,
but a screwdriver would also suffice) is required to make a compartment
otherwise in reach to be "outside the zones". You should still put the
system on an RCD, and the boiler must be room sealed (balanced or fanned
flue). I believe it actually acceptable to have a boiler directly in a
bathroom not enclosed, but not within 40cm of the bath.

Christian.