Which British animal can run faster going uphill than it can going downhill?

Lots of animal could fit the questions it's all "legs related" but since you mentioned " British" I would guess down under "Kangoroo"

I wasn't thinking of a Commonwealth realm. No, this is an animal native to Britain.

Having trouble with this one, but many sites say that hares and bulls have this oddity, so I will guess (based on some searching) that you talking about the Dartmoor hare ?

What's a Dartmoor hare?

The hare is indeed the answer I was looking for. It's certainly true of the European brown hare [Lepus europaeus], the one found all over Northern Europe, including Newmarket (and Dartmoor ), which is what we mean by 'hare' here. It's not likely to encounter steep slopes,living on lowland grassland.Interestingly, its 'cousin', the Mountain hare [Lepus timidus], found in the highlands and mountains of Scotland, has much shorter back legs than the brown hare.It won't have the same problem in running downhill.

I am not even sure it is actually a different species, I answered that way because I loved the website I encountered : Dartmoor Hare

No, it isn't a different species, 'spit. It's the brown hare. The author is merely making a parochial boast, that his brown hares on Dartmoor are tougher than other brown hares. They'd need to be. It would be hard to imagine a less hospitable place than Dartmoor.The grazing is very poor and only very tough animals, such as the Dartmoor pony ( a variety bred for hardiness) can live there. The great heathland is famous for having a high-security Victorian prison (nobody could escape and live long on Dartmoor), for its use by the Army in survival training, and for the number of walkers who die on it, usually from a combination of inadequate equipment and clothing, and exposure.

Awesome info... thanks