As far as your car is concerned, you are required to carry a warning triangle and hi-vis jacket (and use them in the case of breakdown), have a GB sticker or EU style number plates, mask out or otherwise adjust your headlights so they don't dazzle. The old yellow headlight requirement has been dropped.
For the bicycle, you're supposed to have a bell and an i.d. plate, with your name and address, fixed on the bike. Not that I've known anyone to bother. And use cycle lanes where available in preference to the road.
