But, seeing as how the Tudors believed that the King Arthur of legend had been a genuine King of England, the new Arthur would presumably have been crowned Arthur II (not to be confused with Arthur 2, an indescribably bad film starring the late Dudley Moore).
This would mean that, for modern scholars, there would have been a King Arthur II, but not a King Arthur I. Which would have been pretty confusing, if you think about it. Which I have been doing—because I tend to worry about these sorts of things.
Perhaps it's just as well that Arthur, Prince of Wales popped his clogs when he did.
I'm just saying, that's all.