You can ride a horse or a bike on the road. Legally speaking, you cannot ride a bike on the pavement unless it is a shared use path, i.e. a cyclepath. I doubt such legislation would cover riding horses on a shared use path although there are instances, e.g. the trans-pennine trail
Our posts seem to have overlapped, NBT. On the TPT horses are only permitted on bridleways and the parts of the TPT which use the highway. Cycles and pedestrians may use bridleways but cycles are supposed to give way to horses. Even if a bridleway follows the route of a road, (as it does in Charlesworth outside the farm where I keep my horses) horses are still not allowed on the pavement. They must use the actual roadway.
To be very honest I
have broken the law occasionally when a previously quiet road has suddenly become a stock car track in a matter of seconds! Somehow a court appearance and a fine pale into insignificance when an idiot with a death wish is driving straight at me. Drivers are often unaware of the damage that can be done to their car if they run into half a ton of horse!
Yes, I know, I know, there
are idiot horse-riders too. Sensible ones train for and take their Riding Road Safety test and wear hi-viz clothing.