Much like the Queen could have stopped Brexit (and produce an institutional crisis), because it's her job to protect the country above all else. Unfortunately, she didn't.
As reality has proven, she's not allowed to do that job at all. "Having expressed her views, The Queen abides by the advice of her ministers"[0].
What this shows is that there is a power vacuum at the heart of the elective dictatorship that is the UK constitutional system. Actually implementing this mythical "safety valve" would require someone with a proper mandate to exercise the powers of the monarch (vetoing laws, treaties, and wars, for example). That person could be the "Governor General" of the UK, for nominal consistency with the Commonwealth realms, and they would report directly to the Queen, just as the PM does, except they would be elected by the whole country.
To keep them accountable, they could be impeached by a majority vote in both houses of parliament, but removing them would trigger a new Governor General election (with the current holder of that office not eligible). Also, someone could only become a candidate for that role if they received the support of 20% of MPs, which should hopefully prevent a populist sweeping into power. The voting system could be something like the Supplementary Vote[1].
Specifically in the case of Brexit, we could have seen a principled Governor General (perhaps someone like John Major) threaten to resign (during the crisis of May failing to get her Brexit deal through parliament) unless a second referendum was called. An election for Governor General under those circumstances would effectively be a single-issue election on the question of Brexit anyway (without the constituency/gerrymandering problems of FPTP), so I imagine the government would have granted Major's request.
[0] https://web.archive.org/web/20100414023100/http://www.royal....
[a] https://en.wikipedia.org/wiki/List_of_United_States_presiden...