His two younger children would definitely not have been present. Charles would have been at Whitehall Palace, and Elizabeth would have at Coombe Abbey in Warwickshire. The plotters had planned to kidnap or kill Charles in the immediate aftermath of the bombing (they'd originally assumed he'd be present, but revised their plan when they found out otherwise) and to raise a rebellion among the local Catholic population to capture Elizabeth.
The former has a decent chance of succeeding, since one of the plotters (Thomas Percy) was an officer in the palace guard and could probably have gotten Charles out (or at least gotten to him) under the pretense of taking him to safety. The latter was doomed to abject failure: the population they'd been counting on to rise up and help them was mostly loyal to King James and uniformly horrified by the idea of the plot IOTL.
But supposing for the sake of argument that both Charles and Elizabeth were killed somehow, the line of succession to the English throne would have been:
- Arabella Stuart, James's cousin on his father's side (he'd had claims to the English throne through both his parents). She might or might not have been present at the bombing, as she was employed as a lady-in-waiting to the Queen at the time of the plot.
- (Disputed) Edward Seymour, Lord Beauchamp and his children. He probably wouldn't have been present at the bombing (his claim to his title was somewhat irregular, and he wasn't listed in the "state of the peerage" record for that session of Parliament), and even if he were, he had three sons (ages 19, 17, and 15) and three daughters. The sticky part for Seymour and his issue is that as of 1605 he was legally a bastard (his parents had married secretly without the consent of then-Queen Elizabeth, and they had no documentation or witnesses to prove the marriage had actually occurred). That ruling was reversed IOTL and the Seymours officially restored to the line of succession, but not until after 1605.
- Anne Stanley and her sisters. They almost certainly would have survived the bombing, having no titles of their own entitling them to sit in the House of Lords (two of their late father's titles, the Lordship of Mann and the Barony of Strange, were in abeyance between them) and not having any particular positions in court.
Each of the three arguably had a stronger legal claim that James to succeed Queen Elizabeth (Arabella on the grounds that James was a foreigner who couldn't inherit an English title, and Edward and Anne on the grounds that they were descended from Henry VIII's younger sister, whose descendants under the Third Act of Succession (still technically on the books at the time) came ahead of Henry's older sister (from whom both James and Arabella traced their claims)), but none of them had seriously pursued that claim when Queen Elizabeth had died in 1605 and James's claim to the throne had had overwhelming political and popular support. Arabella's next under the same legal theory behind James's claim to the throne, so James winning the 1603 succession cuts strongly in her favor.
My sense is that if Arabella survives, she's Queen, and she promptly marries one of Lord Beauchamp's sons (OTL she married Beauchamp's second son William) to tie together their claims. Anne is unlikely to dispute this, since unlike either Arabella or the Seymours, the Stanleys never seem to have intrigued in favor of the claims to higher positions in the line of succession. If Arabella dies, Beauchamp or his oldest surviving son is probably the heir (they seemed more interested in the throne than the Stanleys ever did, and a man with three teenaged sons had a big advantage over a never-married woman), and they likely try to marry Beauchamp's oldest son to Anne Stanley to reinforce their claim.
There are a bunch of potential heirs after the Seymours and Stanleys, but it's unlikely to come to them. As Thomas Wilson observed in 1600, the crown of England is not likely to fall to the ground for want of heads to claim to wear it.
The Scottish succession is quite a bit messier, since there are no more legitimate Stuarts going several generations back, and since Scottish succession law was quite a bit fuzzier than English. Arabella is one of the plausible legal claimants, but her claim is one of the weaker ones (she's junior to Ludovic on the Lennox line, and she's female, which is a problem since the Scottish throne was arguably subject to a semi-Salic rule). The most likely claimant is Ludovic Stuart, Duke of Lennox (the Lennox Stuarts were descended in legitimate male line from the pre-Royal Stuarts), who has a major figure in James's government and one of the highest nobles in Scotland. But if Ludovic had been present at the bombing (he might or might not have been: he was a member of the Privy Council, but at the time he didn't have any English titles), then the major alternate claimants would be the Earl of Arran (less distant relatives of the Royal Stuarts, but through a female line, and the Earl had been officially recognized as James's heir before James's son had been born, but as of 1605 he'd been confined as a lunatic) and the Earl of Moray (descended from King James V of Scotland along a bastard line on his mother's side, and from another cadet branch of the Stuarts(the Albany Stuarts) on his father's side).
My guess is that Ludovic inherits Scotland if he survives. Otherwise, it's probably between Moray or Arabella, depending on how interested the Scottish nobility is in keeping a personal union with England.