I am pretty sure the Queen herself would not agree to be called "Elizabeth the Great".
I hate quoting The Crown, but there is a scene in that show which is very emblematic of the point I want to make. Elizabeth is talking to someone (probably Philip, but I don't remember exactly) and then says something like (not exact words): "When I became queen, this country was still great. Now the place is falling apart. So much for the second Elizabethian age Winston spoke of".
Fiction aside, the point is that, when Elizabeth ascended the throne, Britain was indeed still a major world power. The Empire was starting to dissolve, but there was still an Empire. Since 1952, the Queen's reign has been one, however, of continuing British decline, a process which may accelerate now even further with Brexit to the point of the UK itself breaking up and England becoming irrelevant in the world stage (or so say the Brits themselves who are fanatical remainers and whose opinion is obviously biased). Of course that is not the Queen's fault, but rather the politicians who have governed the country since the 1950s are the ones to blame. Nevertheless, I think it would be strange to refer to a monarch as "the Great" when her reign coincided with the collapse of her kingdom as a great power.
I would probably go with "Elizabeth the Steadfast", because that is what she is: generally uneducated (as she herself again admits in the fictional The Crown), but dependable, reliable, and, above all, good at being neutral and discreet.