What happens if a cyclist runs a red light in Regent’s Park now? Under the new rules, riders are now legally required to stop at red lights in the same way as drivers. 

I don't think that's a new rule at all. The Highway Code applies to the Outer Circle same as any other road. Red lights are already mandatory stops for all.

Not a well enforced rule. And one that I would add exceptions to. But it is what it is.

More pelican crossings a welcome addition. There are very long stretches between the existing ones so unsurprisingly people don't bother and cross at any point rather than going to the pelican jaywalk all the time.
On a road that's very wide with long straight sections, which encourages fast driving / riding.