They won’t become completely unobtainable. What’s more likely is that some things will increase in price due to (a) import taxes on produce from Europe and (b) a shortage of workers (who are willing to work for the sub-market value wage a lot of immigrant workers accept).
Many of the things that rabbits like best – cabbages especially, also salad leaves – are things that grow well here in Britain year-round. They also aren’t as labour-intensive as things like fruit, so prices shouldn’t be affected too much. You could also grow your own, either in a garden, or on a balcony or window-boxes, or on windowsills (or even under grow-lights!), depending on your living situation. Growing leafy greens and herbs on a small scale is relatively cheap and simple. There are opportunities for foraging, as well, except during winter, as lots of things that grow as weeds are edible for buns.
There is a lot to worry about with Brexit, but I don’t believe your buns will suffer as a result of it. Growing your own and foraging are things that you’ll be able to find information about online, if you need to – probably including from these forums 