Flutter-by
We don't have regional information but I would expect an appointment for an assessment to be issued in 3-5 weeks for an appointment in 1-2 weeks.
A home assessment will depend on the GP letter, it needs to b unambiguous about the requirement for a home assessment, any suggestion that you can attend will likely mean that one will not be offered, so if it says "may have problems..." or "could have.." or "can travel by tax..." then you are going to have problems.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems