The only thing that is clear in my idea is that somehow Sam ends up leaping into the Doctor's companion just as the TARDIS is materializing. Somehow I also hit upon the idea that this would all happen at a Con.
The only other idea I had was that this would somehow involve someone who had -- either deliberately or accidentally -- created some device that drew time travelers. Obviously, our heroes would need to shut down this device before it summoned less helpful travelers.
My semi-practical side -- or at least as anyone could be semi-practical in this situation -- has determined that the companion should come from the period when Sam is leaping: 1952-1999 if I recall correctly. This would put some restrictions on which companion, and thus which Doctor, could be involved. However, if it helped the story or the writing, I could accept a companion who was alive between 1952-1999 even though they were not The Doctor's companion during that part of their life.
I also see some good opportunities for humor: the companion in the waiting room calling for The Doctor, but summoning a doctor. If set at a Con, very surreal humor could come about by the existence of either or both TV shows in the universe in question.
In any case, I would suspect that The Doctor would quickly realize something was wrong with his companion, and would be very suspicious of this stranger. I could also see Sam and Al being very unsure of this eccentric British gentleman and his odd knowledge of time travel. The TARDIS would be of great interest as well.
Two unanswered questions I do have: Would the Doctor be aware of Al's presence, and could Al enter the TARDIS?
As usual when I throw story ideas out like this, I won't make any complaint if they get used by someone. Of course, how can I complain too much about someone stealing my idea for writing a story where they steal several copyrighted characters.
As an aside about crossovers of this nature: In general I would only expect to find such a thing as fanfic. But, despite my believe that it has to have been something I imagined I do recall seeing the X-Men/Star Trek: The Next Generation crossover book for sale at a bookstore once.