Personally once the money and V5 is completed it up to the new owner what they do with it.
It’s been a while but as I recall the seller keeps the V5 which the buyer has signed and dated and is responsible for sending to DVLA? So do that, if as above you can do it online all the better.
It won’t be the first time someone has bought a vehicle without MOT and driven it home, it won’t be the 100th time that month someone has done just that, the penalty is usually a fixed one at £100 and doesn’t mean points. You don’t need a valid MOT to insure a vehicle. If the buyer wants to be legal they just need to book a test, drive to it, spend £50 or whatever on a list of what’s wrong with it and drive it home, but that’s really up to them.