The Doctor would call Ed a Sontaran because of Ed's height issues, Ed would freak out, and then they'd end up somehow banding together to save the day (even though the Doctor would geek out about alchemy actually working).
The Doctor would be fascinated with Ed because he would have heard about Ed and how he was the youngest state alchemist and blah blah blah, and be all excited to get to actually be interacting with Edward Elric. Ed, meanwhile, would be getting very frustrated that for one, this guy is just so damn cheerful, and for two, he knows something I don't and won't TELL ME! Which would end in Ed rambling and yelling at the Doctor, and the Doctor being amused. Then something bad would happen, and the Doctor would run off to fix it, and Ed would end up getting dragged into the mess, just like the queen, and Charles Dickens and Agatha Christie all did.