Some thoughts on this.
One of the reasons that Elizabeth's Court selected James VI of Scotland as her heir was that he was an adult male with a male heir, the fact that he was the King of Scotland and him becoming their King would create a United Britain was just additional gravy as far as they were concerned. The alternate heirs at that point were minor figures in the political arena of Europe, either descended from female lines, or were politically discredited due to the actions of their relatives (like the Greys).
But Elizabeth dying this early opens things up a bit, while Mary of Scots is in hand, she is a prisoner, a political failure in her own Kingdom of Scotland, and a Catholic. Elizabeth's councilors were pretty much solid Protestant at this point, even if in England there was still some hope of a Catholic Revival, if you were ruthless enough to impose such a program.
Right now James VI is a child and in this era children can die very easily, he's under a Scottish Regency, and if he does die then the Scottish Succession becomes a mess since there's a bunch of people with blood claims, but the vast majority of them are from illegitimate bastards of James's grandfather.
The Greys have managed to discredit themselves due to the 'Queen' Jane thing, Katherine's actions with her 'marriage' have left her stained and while she does have sons, sorting out their legitimacy would require the support of Parliament.
Charles Lennox is available in Scotland of course, but his father is the Regent of Scotland and a link like that might be seen as a negative in Elizabeth's court, there does seem to be a preference for a 'home-grown heir' during Elizabeth's reign, something that never happened in RL.
Margaret Clifford, Countess of Derby is a solid choice, her disgrace happened in 1579 (I'm assuming the Date of Death happens in the early 1570's), granted her marriage to Henry Stanley is stormy but she does have two living sons. However the elder son, Ferdinando Stanley winds up being suspected of being a secret Catholic in RL, but in the early 1570's he is his pre-teen years. His brother William Stanley is a child at this point, but in RL he is noted for being a playwright and a possible candidate for being the 'real' William Shakespeare.
But if the Court of Elizabeth desires a male ruler now rather than wait, then Henry Hastings, Earl of Huntingdon is a solid choice. He has Plantagenet blood, he's a Protestant, and while he has no children he is raising the son of his brother to be his heir, so if anything happens there are two other men right behind Henry to take the throne.
In my opinion it's a choice between Margaret Clifford and Henry Hastings, and while Margaret Clifford has more legal weight to her case, Henry Hastings has more advantage due to his gender and a clear line of male heirs behind him. It wouldn't surprise me if commentators in this era would compare this to the dispute between Stephen and Matilda, but in miniature since it will probably play out in the shadows between courtiers and in Parliament, since the loser will not be in a position to get foreign support for an invasion of England to press 'their' rights.