Didn't the British kind of dislike getting dragged into German nonsense Hanover was involved in?
It indeed became more and more a thing, but remember it was originally asked for from Vienna Congress to not only restore the personal union, but as well to give Hanover a royal title. If they wanted to give it up, they could have done so earlier.
See the role of British army in the Electorate, earlier, during the SYW (or the politic of expension).
Not to say it was a major focus, unanimously held by British elites (it was clearly not), but most of the negative opportunities it created.
I could have sworn I'd read about resentment in London over having to clean up after the little corner of Germany they'd been stuck with.
Anyway, I suppose I'll go with my original plan that just involves Henry dithering more after his brother dies (bad weather for travelling, feeling sick, etc.) so the Austrians panic at threat that is really just the foe being indecisive.
That said, a King of France doesn't need to be coronated or sacred to be king of France. The succession at this point is pretty much reputed being automatic.
You could see François d'Alençon warming up the throne, that he attempted to take with the
Malcontents Plot. With an Henry III somehow dithering in Poland, I wonder if the mother queen would be as successful she was IOTL.
The probable alliance between Malcontents/Politiques and parts of the Huguenots (that existed IOTL, prefigurating Henri III and Henri IV policies) would make François in an unstable but maybe favourable position.
Basically : more Henri III is waiting, less likely he would get the French throne.
(And Habsburgs really doesn't have an interest having François getting crowned and sacred)