Just an update, I'm sitting in the Philadelphia airport, killing time before getting a flight straight to Glasgow at 6:45. I emailed the program, who said that they probably wouldn't leave the university until 9:30 or 10 anyway and if I still missed them, there are two students going at 11 once they finish an exam. So I'm fine, I'll get there eventually, and for now I'm just trying to stay awake until my flight goes. On the plus side, I can check "eat a Philly cheese steak in Philadelphia" off of my list of things to do before I die.
So all in all, this could have gone much worse and everything will work out, just not exactly according to plan. But since when has anything ever gone exactly according to plan? :)