If Prince Arthur had succeeded Henry VII instead of dying in 1502, would he have been King Arthur I, or would the legendary King Arthur have figured into the regnal numbering scheme?

I'll extent the answer given there: England's number system only went back to William the Conquerer. For example, the first Plantagenet King Edward (in 1239) was Edward I, his son was Edward II, etc. However, before this, there had been King Edward the Elder (899-924), Edward the Martyr (975-978), and Edward the Confessor (1042-66). So Edward Plantagenet was the 4th official* King Edward of England, but was only known as Edward I, as he was the first Edward after William's invasion. Thus, while we can't answer a hypothetical with certainty, it seems likely that Henry VII's son would have been King Arthur I.

*Edward the Elder was, according to chroniclers, accepted as King across all England, but archeologists have found that York was still producing its own currency, suggesting that he held no real authority over the Danish areas of northern England.