We bought a car from our daughter a few months ago. She forgot to send in the Registration Document straight away. After she had sent it, she received the reminder about tax being due. Meantime, of course, as the new owners we have insured the car in our name. We have not received a new Registration Document yet. She is going to renew the tax online as she has the reminder, is the fact that the insurance is in the name of someone else likely to be a problem.
Can't see why that should present a problem. Reminder in her name, policy still valid, and an mot, that's all you need I think! I have bought cars without tax and got the owners to tax it before collection!