Titles not necessarily constant
The appendage "II" to the queen's name is only correct for England. That is, she was only Elizabeth the Second in England.
In Scotland and the other Commonwealth countries there never was an Elizabeth the First, so she could never be the Second.
When she was crowned, there was a big stushie in Scotland over that naming convention. To the extent that a great number of replacement pillar boxes with the initials E2R were damaged. The result to this day is that there are very few E2R pillar boxes at all in Scotland, in fact there are more with the old George initials.
While we're on the subject, Elizabeth never was the "queen of the UK". That is factually not the case.
In England she was crowned queen of England. Wales as a militarily conquered nation is included in that.
In Scotland she was Queen of Scots. With a separate coronation.
Charles will need to have his separate coronation as King of Scots. Interestingly, the law around that is that the king is king with the acceptance of the people. In theory the Scots people can replace the monarch if they so wish. Being king/queen of the people, not of the country.
A bit off topic: the queen Elizabeth the First (of England) was the one who ordered the chopping off the head of her cousin, Mary Queen of Scots. On the basis that Mary was a threat to Elizabeth's holding of the monarchy.
Last laugh came when Elizabeth died, and Mary's son James who was the Scottish king at the time, got the throne of England as well.