Are academics allowed to own/be part of/found a private company (UK)?
I believe that this is university based. To make a long story short, the university usually owns some form of copyright on your intellectual work and passes it to you when you publish. If any of said copyright is used for commercial purposes then you need to make a deal with them on a case by case basis, though there are standardised contracts. University will have a office devoted to this. If the company is unrelated to your field/work than there's no need for such arrangements, but you still might want to double check with them.

This is my understanding having seen a colleague going through some of this (and eventually get told there was no need to make arrangements, essentially if is not something you devised or developed while working its ok)