Will the public’s questioning of the relevance of the monarchy increase when the current queen dies?

I'm pretty sure that's exactly what will happen when the current queen dies.

It should but it won't

Outside of Yahoo answers the vast majority support having a monarchy.

Yes. It's already being questioned as being out of date and a waste of money.

Yes, 90% of the population will want the Monarchy abolished rather than let Charles and Camilla loose on the nation.

As a French royal, I know that "people" (not subjects anymore), still honor the Royals in any country.

Yes there can be no doubt about that but, not enough to make ANY difference to the status quo.

I doubt it. Most people will mourn/miss the Queen, and then get on with their daily lives.

No. There are people questioning its relevance now, but there still is a majority who prefer the constitutional monarchy which means no change to government structure.

Almost bound to, orchestrated by some sections of the media, but it won't change the fact that about 90% of the population is either in favor of the monarchy or doesn't want any more upset to the constitution.