The BBC said he had chosen Charles III a few hours ago.I associate "Elizabethan" with Elizabeth I.
If the reign of the new monarch is going to have a special name at all, it will need to be distinctive from what comes before and after. Hard to say that this will be the case before it happens (though considering that Charles is already 73, it's likely only going to last a fraction of the time that his mother's reign lasted).
And some past eras have been distinctive because of change, not constancy (e.g. the Year of the Four Emperors), so the upcoming era might be named for something other than a specific monarch.
All that aside, if we're coming into an era that will be named for this specific new monarch, it will probably be influenced heavily by the name he takes as king, which AFAIK hasn't been announced yet (but there are good reasons why he might avoid being "Charles III" officially).
I have heard us called 'Elizabethans' quite a few times, tbf; possibly as the age was so long.