If either walked into my workplace I'd treat them respectfully... but I'd not hide that I dont believe in monarchies if asked. In a polite manner of course.
From my point of view, "Elizabeth Windsor" is deserving of respect. This, in my current definition, is something different from what you label 'respect', which I would term 'be polite towards'.
She's a deserving symbol of the nation who has spent most of her life in public service, among many other things. Her title lends her more gravitas, has greater implications, than "old lady who I like because she's healthy." It means more, and if you don't feel it personally, you have to admit that a great many people do; I would even suggest most.
The following sums it up:
I would be polite to Elizabeth Windsor, as I would be polite to anyone who seems deserving of it.
I would respect the Queen.
That being said, there are some who are not deserving of being that symbol, and whose character is not suited to it. Many see Trump as just one such.
Alas, we must separate person from position and simply be glad that there do exist those who are worthy of their place.
If Elizabeth Windsor were a disreputable lay about who stole candy from kids, I'd sum my argument up as follows:
I would not be polite to Elizabeth Windsor, as I would not be polite to anyone not deserving of it.
I would respect the Queen.
The issue being that that respect entails politeness unless in private, but ah well. Even a public meeting isn't hugely likely.