It's hard to imagine anyone mounting a significant challenge to the succession of King James of Scots. Primogeniture, religion and the political expediency of neutralizing the Scottish threat are all in his favour. With the reformation driving a wedge between England's traditional go-to alliances in France and Spain, a personal union with Scotland (nixing the Auld Alliance) and an already-made dynastic alliance with Denmark-Norway (strengthening England's hand in the North Sea immeasurably versus potential Spanish challenges coming from the Low Countries) are both highly advantageous to national security.
As for the potential rivals to his succession, the Lady Arabella Stuart was favoured by primogeniture and the traditional preference for an heir born in England. As of 1589 she was a girl of 14, unmarried and inexperienced. An appealing candidate for those who might hope to play King (or rather, Queen)maker and dominate government to personal profit, but less than appealing to those Protestants who found themselves ideologically opposed to the dominion of women.
Edward Seymour had the double boons of being an adult male (26 years old) and English-born, but primogeniture stood against him as did the imperfections and questions regarding his birth. As was seen in Mary and Elizabeth, a dubious origin and even outright bastardy were not complete impediments to royal succession in Tudor England, but unlike them Edward did not have the law (by way of an Act of Parliament) or popular and council support clearly on his side. In his favour he has two young sons, English- and true-born. It's worth noting he carried the courtesy title of Beauchamp as only his father's legitimate heir would have been entitled to, and following the Restoration the family's titles were all restored to his son (William Seymour), who became the 2nd Duke. Thus, it wasn't impossible to regard him as legitimate, as he certainly was when it came to the succession of the Earldom of Hertford and other Seymour titles and property. Still, making him King is a tall order. Also in his favour is the fact that he was at court during this period, and could perhaps move faster than James.