So the queen is alleged to favour Brexit. Whether or not she does side with the Outs doesn’t matter. As monarch, she is not supposed to have any political opinions at all. Or, to be more specific, Elizabeth Windsor can think what she likes; Elizabeth II on the other hand must remain neutral and silent on any such issue. That is what constitutional monarchy means.
We the people have reminded the monarchy about this on several occasions, notably in 1215, 1649, 1688-1689, 1834 and 1936. As a recent play pointed out, it will be a rule that Charles III should bear in mind from the very beginning of his reign. Not too difficult, surely.