I think if it were me I'd go and buy it, drive it straight home untaxed but insured and MOTd having picked it up on a Sunday or something, and, if stopped, explain, showing documentary evidence, that I'm waiting for a cover note in the post and I couldn't have taxed it on a Sunday anyway.
Trouble is, with the "policing by post" system we have these days, you're liable to get caught on camera and have a bill for whatever the fine is dispatched by post with no opportunity to reason with anyone nor any grace period.

Almost every used car has the tax cashed in so the buyer can't take posession of it legally when it's purchased. It was the same when I bought my car. I traded in my old car so how was I going to get home in a new car without tax? In the event I drove it to the nearest post office and taxed it. Luckily I had the foresight to get insurance documents organised beforehand but I still drove it untaxed to the post office.
Kevin