If a designer has produced a set of full drawings and is paid for them, and the client subsequently sells the site on, is the new owner legally able to use the drawings to build on the site?

This is predicated on copyright. If the terms of engagement explicitly state that the licence is applicable to the client only, then a future owner of a site will have no licence to use the design. If there is no mention of this issue, then it will come down to a court decision, but it is legally likely that the court would rule that a future purchaser inherits the licence. Occasionally future purchasers may be barred from obtaining the licence if a clause explicitly states that transfer is not permitted without the designer’s permission.