Lynx are native to Scotland, I don't believe that rabbits are, they are usually considered to have been introduced by the Normans in the 12th century. According to the wikipedia article the lynx appears to have a pretty varied diet

In the case of eurasian lynx replace 'rabbit' by 'hare' and you got it. They eat also a lot of roe deer and maybe a few (very young) red deer.

Why wouldn't rabbits be native? It's not so long ago that Great Britain was connected to mainland Europe, so I'd think that most of the same animals would be present both sides of the North Sea.

European Rabbits are native to the iberian peninsula and southern France, they were introduced to the British Isles either by the Romans or the Normans. There's more information about this on the Wikipedia article https://en.m.wikipedia.org/wiki/European_rabbit