Normally it should take less than a week but things are not normal so my advice would be to chase the DWP every day or two until you receive the Enhanced rate, the Mobility may require an assessment and you may have to explain why you qualify.
You should be entitled to the LCWRA element of UC again as a rapid claim but again you will need to chase, the DS1500 should remove the five week wait for a payment with UC.
PS - Your post was not lost, they only appear in the forum when approved by a Moderator.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems