I'm not sure a change of monarch would make any difference. Whoever it is, they can't say anything, can they?
I think it would be an emotional and existential shock for the "U"K. Let's just start with the fact that it's been debated - quite openly - that quite a few people like the idea of Charles abdicating so Wills can be King. Add to that the fact that there is a republican sympathy in society, and a change of Monarch might amplify that. Especially if it seems "the will of the people" for William V seems to be running high.
Then there's the cultural shock - there are more people alive that have only known one Monarch in their lives than can remember the last change. Myself, DW for a start.
And the whole thrust of Brexit has been chained to that "Rule Britannia"/"God Save the Queen" jingoism.
Bear in mind, Charles will be coming "to power" completely cold. He is frozen out of the briefings the Queen gets from her government. Briefings which can be two-way. They're never minuted, and no one else is ever present. We have no idea if the Queen has suggested or advised, let alone what it might have been.
Given that last, I have wondered (on this very forum) if Theresa May has been getting sage advice from the one source that no one can nobble.