As the work will cost £300, I just wanted to double check here with
the experts, that an moisture triggered extractor fan in one room,
will sort out the problem for the whole house.

It may or may not. Assuming that penetrating and rising damp is definitely
not a problem, then the basic problem is that the downstairs entrance hall
has cold, wet air. The extractor fan will hopefully reduce the moisture, but
the real difference between that part of the flat and the rest is that there
is no heating.

As the wet air tumbles down the stairs, it gets progressively colder. Cold
air can't hold moisture, so it condenses out onto the walls.

It is certainly a very good idea to have a kitchen extractor anyway. I would
be inclined to see if it helped before determining if another cause is
indicated, as you should have one anyway.

Other things you should be checking are that you do not have flueless gas
heating and that you don't dry clothes by putting on a radiator, or use a
vented tumble dryer with no vent.

Christian.