Correct me if I'm mistaken (My wife does..) but the original question was about the legal position. Which is easily answered & has been.
If you're a student on a driving course, they will insist upon helmets from an insurance & communication perspective.
There may be confusion for future owners from a driving licence angle, but the car is classed as that - a car. Seatbelts are as mandatory as any other car use.
Eye protection is down to personal preference & isn't mandatory on one, two, three or more wheels. Recommended, but not compulsory. If you have eye protection fitted to a safety helmet it must conform to standards, but it's ok to wear Raybans if you so choose.
Please don't lets get into the usual "My mate would be dead if he'd worn a seat belt". There's an exception to every circumstance, but the laws are clear.
