You could evaluate the risk exposure of your UK bank reading this post and this other old one. They basically say that UK bank exposure to Greece is less than 6 billions pounds (BOE data), so there is no reason to be worried now.
The main issue of this crisis is not the Greek exit from the Euro on its own (it seems to be considered almost a fact by CITI, and by MS at 35% probability, Profumo ex CEO of UNICREDIT, says the possibility are more than 50%) – the main issue is that other countries like Italy and Spain might follow the same fate.
If they do, the exposure of many foreign banks (including the UK ones) to their debts is not negligible (191,80 billions pounds for UK banks) moreover other EU banks (even the German ones) exposed to Italy and to Spain will suffer too, and this suffering will be translated into more suffering for UK banks exposed also to Germany and to France.
That's why you read Euro doom articles like this one from Paul Krugman (who won a Nobel Memorial Prize in Economics.)