Very few few FIs in the UK are employed so they will not have a contract of employment! The few that are employed should have one, but I'll bet most do not.
I have never heard of anyone being vetted, no doubt you would have to pay for this out of your non existent earnings so I guess youngsters will have to go some place else. I would certainly never get tied up with such bureaucratic nonsense.
Insurance is interesting, in the main part the student will be covered by the club but you as the instructor probably won't be. What insurance are you referring to? Liability insurance in case someone sues you, or cover in case you have an accident.
Liability insurance can be obtained
here The Club should be able to include the Pilot (instructor) on their insurance but it probably won't cover liability.