All King Air aircraft are single crew. If the aircraft is operated under a UK AOC for public transport then it will require 2 crew, both type rated and as you say, both log the time. This is only because the CAA require it, so under any other operation, in other words not an AOC operation, it will be single crew and anyone occupying the right seat, type rated or not, cannot log any of the flight time. If the aircraft is certified to be flown from the right seat then the smae would apply, only one pilot logs the time.
Many King Airs in the UK are private owned and are flown single crew, even by commercial licenced pilots, but it is a private operation.