Have a peek here.
https://www.gov.uk/vehicle-registration/overview
I guess the form you need is V55/5. I think you can buy indicator kits to put on most quads, there's likely a generic one too. I imagine it's already got lights and a horn. The registering process is £55. By the looks of things you need to supply some documents to prove you are who you say you are, you know, passport, driving licence, a kidney, etc. You'll also need the cost of the road tax (unless registering it agriculturally).
However, as this is basically an administrative task, were I you, I'd suggest that the H&S gestapo that are so keen on this could perhaps be best placed to do it themselves.