Are academics allowed to own/be part of/found a private company (UK)? I have seen some professors being members or even founders of one or more private companies. Many times in a similar topic as the one they teach/research.

What is the legal status behind this in the UK? Is it universal (controlled by state laws and rules) or contract/university specific (to be negotiated with the employer)?
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)