I think Brexit, were it to unfold, would be extremely bad for the UK banks. The UK economy would take a hit equal to many tens of billions of pounds, which would, in turn, result in significant job losses and numerous business failures.
There is no way that the UK banking sector is adequately provisioned for such an scenario.
At any rate, I have my doubts about Britain exiting; the stake are simply too high. And a number of high profile monetary managers and economists have gone on the public record, arguing as much, and quite prominently so, in recent weeks.
I think Brexit will end up being something like Y2K (not sure if you are old enough to recall the months leading up to 1 January 2000): made for great, catchy and dramatic media headlines - selling lots of newspapers in the process - but turned out ultimately to be little more than a damp squib.