If the current car isn't insured when you want to sell it, unless the test driver puts it on their policy (which their insurers probably wont allow for test drive purposes) they can't drive it...that'd put me off more than no tax...
If it's only a week or 2, can't you just get a new policy on the new car, let the old one run out and hope it sells in time? If it's a question of NCD Transfer, most insurers give you a couple of weeks to send it in, then send a letter reminding you to send it off anyway. Then the current car can be put on the new policy as a temp vehicle for a week or so (my insurers allowed 4 weeks maximum for my own second car for the purpose of keeping it on the road to sell) then people can test drive it on their insurance TP only.