Having a potential male heir available would likely greatly increase the power of the Catholic faction in England. This *James VI is born in 1542, so when Mary Tudor died he would be 16, virtually of age. I doubt Mary would ever disown Elizabeth for *James, but who knows what the conservative nobility would do after she was gone? If the English Catholics could pull together behind him, there might never be a Queen Elizabeth; or, a shotgun (blunderbus?) marriage could be arranged (although this might be hard to pull off; weren't these two first cousins? If so, they'd need a papal dispensation, and who know if that would be forthcoming?).
I'm not sure this would even work to keep England Catholic, though. Protestantism was pretty entrenched in London, and Edward VI's visitors had already trashed a lot of the medieval iconongraphy that connected people to the old faith. And even nobles and gentry with Roman sympathies weren't about to return the property they, um, acquired following the dissolution of the monasteries; I've gathered this was a sore spot between the English government and the papacy that Mary never figured out how to negotiate. Continental-style religious civil war seems likely. If young King James inclines towards chivalric traditions, he might solve the divisions the way Henry V did--by invading France.
This POD would, incidentally, rob us of one of the better titles in English literature; John Knox would never have cause to write "The First Blast of the Trumpet Against the Monstrous Regiment of Women."