If the conspiritors get Prince Henry and Prince Charles as well as King James, and if they succeed in kidnapping Princess Elizabeth, then there's a significant potential succession crisis in both England and Scotland. Elizabeth is unquestionably James's heir once the death of both Princes is confirmed, but she's being held hostage by a band of traitors. They may kill her when it becomes clear their situation is hopeless, or they may be able to spirit her away overseas (probably to Spain or France, either of which may be interested in using her as the figurehead of a Catholic government-in-exile); in either case, another heir is needed.
For potential candidates, search the forums for discussions of alternative heirs to Elizabeth I, as this is only two years later. Two of the candidates would have been killed in the explosion: Edward Seymour is in the House of Lords, so he'd be killed and his claim passes to his son William, and likewise with Henry Hastings Earl of Huntingdon, whose claim passes to his sister Catherine (wife of then Sir Philip Stanhope, who later became 1st Earl of Chesterfield).
The English candidates are Lady Anne Stanley, Lady Arabella Stuart, William Seymour Earl of Herford, and Lady Catherine Hastings. There's also some descendants of Henry VIII's bastards running around, who may try to put in a word for themselves (Colonel Sir Edmund Carey (who would become Baron Hunsdon upon the death of his brother), descendant of Mary Boleyn, is probably the geneologically senior illegitimate descendant of Henry VIII once you remove people who would have been killed in the explosion).
The Scottish candidates are Lady Arabella Stuart, Ludovic Stewart Duke of Lennox, and whichever member of the House of Arran is best able to stand in as regent for the confined lunatic James Hamiliton 3rd Earl of Arran.
My money is on Anne Stanley in England (best legal claim and nobody really has as stronger political position) and Ludovic Stewart in Scotland (by far the strongest political position). Unless Arabella Stuart manages to claim both thrones (very unlikely in Scotland, where she was not even able to claim her father's titles because she was seen as too English), personal union between England and Scotland is dead for the time being.