The obvious difference would regarding the Edwards. Honestly, one could argue that the regnal numbering should really start with Alfred the Great, but if we're starting with Æthelstan, then Edward the Martyr is known as Edward I and Edward the Confessor is Edward II, meaning my dude Longshanks becomes Edward III, while the last Edward is Edward X. In effect, Edward pretty much becomes the English equivalent of Louis, Alfonso, or Constantine in the sheer number of monarchs with the same name per kingdom. If you want to count back to Alfred the Great, then everybody moves up an ordinal and we are up to Edward XI.
Not sure about what the possible political or cultural ramifications would be however. I suppose it would depend on Henry III actually, since he revived the name Edward in honor of his hero Edward the Confessor. Perhaps in order to affirm his dynasty's Englishness, he decides to have work done on a project to illustrate how far his bloodlines truly go, all the way back either Æthelstan or Alfred. Notably, he probably skips over all the Danish kings and over Harold Godwinson, who was in direct opposition to Henry's direct progenitor William the Conqueror. Henry had clear knowledge of English history, he wouldn't name his sons what he named them if he didn't.
