Much depends on what sort of man Henry/Enrique turns out to be, and how much he listens to his father. Phillip didn't believe a personal union between Spain and England was possible; even if both are Catholic, the English hatred of the Spanish and predilection for piracy make it untenable. If Mary predeceases Philip as OTL, Philip will force his son to sign a treaty holding the crowns separate forever (which will delight Parliament). So the fun begins when Philip dies and Henry presumably leaves London for Madrid.
The best legal claimant is, as noted, Mary Queen of Scots, whom the Valois will back because the idea of a Habsburg England would seriously frighten them. However, Henry will probably name some other heir and get Parliament to ratify it - possibly some minor Habsburg, possibly the Duke of Norfolk. In either case, though, you now have two competing claimants for the throne, both with a major foreign power backing them. Let the English War of Succession begin.
If Henry attempts to rule both Kingdoms by tearing up the treaty he made with his father, well then, let some different games begin; he'll still physically reside in Madrid and try to install a viceroy in London, said Viceroy will be first ignored and then murdered, and we're off to the races.