New owners should remember, it is extremely bad form to organise any part of the move until missives are concluded [contracts are exchanged outwith Scotland], because if the deal falls through, this can cause problems for continuing owners.
Yes, nothing should be done until contracts are exchanged. There will also be a gap of 14 days (10 working days) anyway, because the 'losing' ISP sends a letter to the 'old' occupants, saying 'sorry you're leaving us' as an anti- slamming measure.
It's important that the old occupants ignore the letter, to allow the process to complete without a glitch.
Typically, the gap between exchange of contracts and completion is 14 days anyway, but if it's less than that, then there will be a gap in the service for the new owners