Th Queen is all for the E.U. She will never do it.
Correct - constitutionally the position is quite simple. The will of the people has been expressed and Parliament is obliged by constitutional convention to give effect to the will of the people. It is technically true that the referendum is non-binding because of the issue of Parliamentary sovereignty - no Parliament can be bound to any decision (part of the reason for voting to leave the EU is because that was starting to be at risk of not being the case anymore) but in the British system, constitutional convention is just as important as constitutional law. The Queen, as guardian of the constitution, would be obliged to step in to ensure the conventions are followed.
People seem to think the Queen is nothing but a symbol and rubber stamp - she isn't. She still has all the same powers that Henry VIII had - the difference is that if she used most of them without an extremely good reason, Parliament would dissolve the Monarchy and the people would support Parliament in doing so - the Monarchy remains popular only while it acts within the restraints of convention. But by the same token, if Parliament violates those conventions, or if a constitutional crisis develops, the people would expect the Queen to step in.
The Monarch has used their powers even quite recently. The Queen has only ever had to make use of them once during her reign (in 1963 when she appointed a Prime Minister without a general election or Parliamentary election) but they still exist. Her father used them in more dramatic fashion - Britain was meant to have a General Election in 1940. It didn't because the King decided that an election during war time would be dangerous and distracting. He got the approval of the Prime Minister and Leader of the Opposition for this, but it was his decision. These powers aren't back in the distant past - they still exist to use in an emergency.
If the current British government decided not to go ahead with Brexit, the Queen would intervene. She would first of all warn the Prime Minister and in most cases, the Prime Minister would probably heed that warning and things would be back on track. If he refused, she would first of all look at Parliament to see if another Prime Minister would be able to accomplish the task, and if so would appoint him or her to do so. If that wasn't possible, she would dissolve Parliament so a new general election could be held specifically to elect a government that would give effect to the referendum - and in such a situation, the British people could be expected to vote to deliver that - even many who voted to remain would be so outraged that Parliament was ignoring a referendum - ignoring the will of the people - that they would add their vote to the even more outraged Leave contingent.
This isn't all theoretical - by convention, a good Prime Minister and a good government does not put the Queen in the position of having to use her powers, and in Britain, most of the time, Prime Ministers do do the right thing in this regard (most recently, Gordon Brown in 2010 who resigned as Prime Minister when the Queen intimated to him that it was time to go - she didn't have to intervene because he resigned at that point, knowing that she was constitutionally correct (note - that he had done nothing wrong at all in trying to stay in office to that point - he had every right to do so - arguably the duty to do so, until she called time. He followed the rules and conventions impeccably). But in the Commonwealth Realms (those other nations that retain the Queen as Head of State such as Australia, Canada, New Zealand, etc) governments have not always been as careful to avoid intervention and so Governors and Governors-General as the Queen's representative have had to step in a number of times and use the powers that still exist. Here in Australia, it happened particularly dramatically in 1975 when the Governor-General dismissed the entire government after it became clear it was about to start acting illegally to stay in office, but a much more low key occurrence was in the state of Tasmania in 2010 when they elected a hung Parliament (10 Labor MPs, 10 Liberal MPs, 5 Green MPs) and the Labor Premier immediately tried to hand over control to the Liberals (the Premier took the view that even though they were tied on the number of seats, as the Liberals had received more overall votes they should be given the chance to govern) - the state Governor informed him that as the incumbent, he had a duty to at least try and stay in office, no matter how honourable his intentions.
I’ve tried to think of an analogy to the US system that might help people understand this. I think this one works.
The idea that the Brexit referendum can be ignored by Parliament can be roughly equated to the idea that in the US, the Electoral College could choose to elect somebody as President contrary to the votes at the November general election. Technically and constitutionally it’s true - practically speaking it will never happen because it would be an outrageous violation of accepted practice that the general public simply would not tolerate.
You may get a few MPs who go against the vote, in the same way that the US has occasional faithless electors, but the number will not be enough to change the result.