IIRC the early part of his reign was very unstable, and I would think it quite likely that the male contender would seize the throne, rather than see a queen succeed a queen.
Best Regards
Grey Wolf
So basically the Earl of Arran becomes King?
I've started a similar thread sometime ago. It might help the discussions here: https://www.alternatehistory.com/discussion/showthread.php?t=143791
At first, the Earl of Moray would probably claim the throne in his own right (as a bastard son of James V) rather than as regent to Mary's infant child as per OTL. Maybe he manages to hold the throne, or maybe he gets overthrown by either the Earl of Arran or the Earl of Lennox.
Being a bastard doesnt he need Papal dispensation in order to succeed?
Why wouldn't the Scottish nobles accept an infant female as they did with the young James IOTL?
As for England, Elizabeth might well be more inclined to acknowledge young native-born Arabella as her heiress.
ITTL, Elizabeth might be more inclined to support someone from the Lennox line as her heir than OTL, but there will be a ton of butterflies. *Arabella might be born earlier or later or not at all, or be born male, and her father might survive long enough for her to be raised in Scotland.
Also ITTL Elizabeth might more willing to arrange Catherine Grey's marriage and declare her as heir.