McCoy forsakes the medical field and becomes Gandalf's disciple in the mystical arts. People ask him for treatment and he shouts "I'm a magician, not a doctor!"
Kirk scolds Aragorn for being a totalitarian dictator and they have a really slow fistfight.
Spock spends all his time playing chess with Sauron.
With Chekhov's help, a team of hobbits sneaks aboard the Enterprise and steals a food replicator. Diabetes rates in the Shire skyrocket.
Scott marries a dwarven brewer's daughter, moves to Bree, and opens an inn to rival The Prancing Pony. Soon the town is embroiled in a Yojimboesque blood feud where travellers are forced to choose sides.
Forced out of Gondor by rampant racism, Uhura defects to the Southrons and becomes a warrior queen.
Sulu fits in with the elves just fine.