What about Henry Frederick?
Dead, as heir to the throne he was planned to be in the building at the time. The plotters planned to grab Elizabeth but as others have said her reaction would have been anger and defiance, they have just murdered her family after all.
Charles wasn't expected to attend, he was a bit of the unfavorite at the time, and there was no real need for the spare to show up (things changed after his brother dropped dead).
I've seen some suggestions the Scots might have scooped him up, scarpered over the border and taken back their independence while leaving Elizabeth for England, but it's hard to know what's really going to happen. Elizabeth does seem to have been a bit more sensible than her brother so she might have done better if she ended up on the throne plus her name would invoke Gloriana in her subjects. In contrast James really wasn't that popular at the time.
I hate to use 9/11 comparisons but he was a bit of GWB, not terribly popular and his policies (full unification, mucking with the quiet consensus about religion) sure weren't. Meanwhile scotland's feeling a bit hard done by with the King having decamped to a more powerful throne. The Catholics really aren't happy having thought James would let up on the persecution and found he...well didn't.
Then came the bomb plot, everybody rallies around the lawful King against the outside enemy, the Catholics duck their heads and stay there, and anyone else with ideas gets a graphic example of what happens if you try to knock the King off.
However if James is dead he won't be raising Charles as he did in OTL, he'll be raised to be a very pious protestant, and to rely on the advice of his ministers that probably lets the constitution as it was stand for at least a few generations longer. One bad King would probably still convince Parliament to flex it's muscles as OTL but it might put things back into the eighteenth century.
A failed coup (and make no mistake it would have failed) and alot of dead (James feared the bomb could have killed 30,000 gathered for the state opening) is very bad for the Catholics. There will be a pogrom, and Catholics will have to pronounce their loyalty even more loudly than they did in OTL, probably by converting.
On the foreign front, Spain and France both want peace with England so they can focus on the continent. Peace with Spain is only about a year old so alot of people will probably suspect they were involved, but after twenty years of inconclusive war the last thing a decapitated England will want is a fight. Once it becomes clear alot of the plotters have connections to English Catholic mercenaries fighting for Spain the Netherlands, Spain likely has to make a few concessions. Probably the Merc's are kicked out and Spain pronounces it's "Prefound shock and disgust at this evil act on the person of a King," (or some such).
Watching the BBC show about the plot last night it does seem the plotters asked a few Spanish contacts for help, and the reaction was pretty much "Are you nuts!" because the Spanish wanted peace. Probably a "Wasn't us this time," from the Spanish would be looked on with askance but after a decapitation strike England's in no state to pick another fight. Things might get cold for a bit and Spain probably stops backing any Jesuit efforts (if they haven't already), but peace should hold.
Bonfire night probably wouldn't have any fireworks but would instead be a night of mourning and ritual anger, I expect relations with the Catholics would improve eventually so by now it's probably a forgotten old shame like the anti catholic slurs that used to be carved on the Fire of London monument. In contrast the act itself is probably still remembered as the most horrific terrorist attack in history.