So, as it was tight in the box around the main power input, I decided to build a smaller plastic box with an accessible fuse and an on/off switch, the same type of plug you linked to me, so it's easy to access and replace the fuse if needed. Hopefully I won't have to do that!
So in the idea for a next design:
-1 main fuse on the live wire to take care of possible safety issues and cut the power going into the device.
-1 post-transformer fuse per transformer to prevent each circuit from a possible issue coming from the transformer
Each of these fuses of course with a different amperage, depending of the power of each circuit.
The only protection is a thermal fuse. Upstream of that, the usual circuit breaker would be 32 A in the UK.
Of course, there really isn't much chance of a short between the plug and the transformer, as there is only a couple of centimeters of wire at most, inside a welded plastic housing.
Which is perfectly fine - the thermal fuse is all that's needed - and is almost certainly a non-replaceable one internal to the transformer. As such it meets all UK legislation. As I suggested above, it's mainly there to prevent fires.