I put my plate on a car last week (Tuesday) at the local DVLA office, new V5 came through the post yesterday - just to give you an idea.
RR, no need for new MOT's when your talking about cars with MOT's. DVLA just re-print the certificate with the new/old reg on it and there is a note on it saying 're-issued at DVLA office bla bla bla' on it.