So what's the environmental differences between a cyclist, a horse, some walkers or doggists leaving little blue poo bags behind? Why allow horses and not cyclists? The difference seems a little oblique and a tad elitist to me. Could you explain your reasoning? I don't cycle there, and never will, but it does seem that if a few [possibly rich and well connected] horse riders are allowed permits, then why not a few cyclists as well? How much damage to the environment there does one cyclist do compared with, say, half a ton of horse shod with effectively four spades on their feet? Seems like the Ashtead cyclists have been a bit dormant in their reasoning?
Well there are several differences but first of all I'd start by saying that yes horses do cause an impact and if you were starting from a position of no access it would be much harder to justify allowing them onto the forest than the current position of justifying their continued use of the forest.
How much damage does a horse do compared to a mountain bike? To be honest I'm not sure but they behave in different ways. If I'm on the forest on a horse I might go around a 5 mile circuit once, a mountain bike might go round the same quarter mile circuit repeatedly and so it will cause more intensive but locallised damage.
There's also a point of enforcement. One might allow as a compromise moutain bikes on set routes or only if they pay for access in the same way as horse riders but you can pretty much guarantee that people would not make the distinction and there would be cyclists on all areas of the forest many of them without paying.
Another problem is one of making it seem that vehicle access is allowed. How long would it be after seeing mountain bikes on the forest that we see motorbikes on the forest. Maybe that wouldn't be the intention but alowing one would be seen by some as a precedent for the other.
There is also the point, which might sound elitist but is practical, that horse riders pay for their access which offsets the damage they cause. So by that token if cyclists were to be allowed they should pay.
So all in all I'd say it should be avoided.
... and just so you know my position. I ride bicycles, motorbikes and horses, and regularly walk on the forest (oh and I drive a cars as well). I regard bicycles as vehicles, they shouldn't be on pavements nor on the forest. Horses (by virtue of having legs) fall in the same category as pedestrians and have access both on and off road. Mountain bikes are acceptable if someone has private land that they've decided doesn't matter if it's damaged.