After some thought, I have concluded that short of replacing Ike in the white house, you could have Harold MacMillan call bullshit on Ike's threat, or have someone other than Macmillan as the Chancellor of the exchequer, or have Anthony Eden disregard the advice he gets from MacMillan, which isn't entirely out of the question, Eden was far from being one of the best PM's the UK has had.
However once you get passed that there is the problem that the islands are now going to have a serious economic crisis on their hands. Add to that the fact that there was an oil embargo on France and Britain as a result of the crisis, and both nations are going to have a hard time sustaining their economies, much less large scale military budgets. So even if britain does tell the US to screw off, shortly after the dust settles I cans see them come crawling back for assistance in short order.