Of course, public opinion will be considered, to be sure, but that will be measured quietly by polls, not through a referendum. I agree with you Emperor of the Dominions, there will be somewhat of a tricky PR situation if Camilla assumes the title Queen (given Charles' prior statements that Princess Consort will be used instead).
But there are many other critical issues on the table than the current whim of the British public. Don't forget, the Queen is not only Queen of the UK, but also Queen of 15 other realms, and Head of the Commonwealth, and Head of the Church of England, and many many other legal roles.
I believe the most likely scenario is that the palace & government will keep their eyes on polls over the years to see which way the wind is blowing - in the UK and the Commonwealth Realms, and the Church of England. When a new reign appears likely on the horizon, the government, palace and other involved parties will open quiet / private discussions about a wide range of issues related to the new reign (from the Crown Estate and Civil List agreements, to the coronation, to the Church of England establishment, Camilla's title, etc.).
In the end, whomever is Prime Minister at that time will have to advise Charles, after having considered the global perspective. You can't have Camilla as "Princess Consort" of the UK, if Canada or Australia don't agree and she is Queen of those realms (or the reverse!). But if Australia or Canada have become republics in the meantime, that won't matter as much. If Charles is going to succeed his mother (or not) as Head of the Commonwealth, that might impact the PMs advice too.
Finally, I don't think any prime minister wants to even tackle this thorny issue until absolutely necessary, hoping to instead pass it on to his successor to handle. So no final decision will be made until it has to be made.