We know for obvious reasons, the Star Trek franchise would not be a reality in the history of the Trek universe.
But what about other science fiction franchises?
Would you ever hear Starfleet personnel discussing the merits of Babylon-5 as opposed to new Battlestar: Galactica?
Or ever hear people at a Starbase debating whether X-Files was better than StarGate SG:1?
Diane Duane's books make it pretty clear Uhura is a Doctor Who fan (she's converting a Tom Baker episode to 3-d hologram in My Enemy My Ally, and offering to trade episodes on the Enterprise's bulletin board in Spock's World).