If you have an Israeli stamp in your passport, there may be a problem. The trick is, when entering Israel, to ask the Immigration Officer not to put a stamp in your passport. In my experience, they will do this without question and will put it on a separate paper instead. They understand why you are asking.
This is what the Foreign and Commonwealth Office web site says about visa requirements for the United Arab Emirates:
"British citizens do not require a visa for the UAE but if you plan to stay longer than 60 days, contact the UAE Immigration officials on arrival."