( Who's Who? )
( What's what? )
When you write a crossover fic like Fringes:One of Those Days, cannon takes a beating. The fic is set roughly in the first season of Fringe, before the discovery of the other universe and before Peter and Olivia began their relationship. That explains why Peter is so flirty with Uhura.
Uhura, as noted above, is from Star Trek: The Original Series. Not NuTrek. How did she become one of the Doctor's companions? Her backstory is sort of tragic. When the Star Trek timeline was jiggered by Nero & Spock, the F:OTD Uhura was shook loose from time and discovered by the Doctor. She is basically a living, breathing paradox.
Tony Newman's history is just as tragic. At the end of The Time Tunnel, Newman and his friend, Doug Phillips, were stranded on the S.S. Titanic, at the same time and place of their first jaunt using the Time Tunnel. The two became trapped in a closed temporal loop. Trapped on the doomed ship, untouched by time, Newman spent forty years subjectively working out complex multidimensional equations in his head to find a way out. His efforts attracted the attention of the Doctor who rescued him. The fate of Doug Phillips remains unknown, but Tony is a bit 'fragile' in F:OTD, suffering from bouts of severe agoraphobia. I imagine he's not too fond of cruises either.
UNIT, in F:OTD, has become the dominant international organization responding to extragovernmental threats and general wierdness. As such they have absorbed a number of other organizations into themselves, including U.N.C.L.E. This explains how Napoleon Solo is working for UNIT as head of their North American Division.
Speaking of Napoleon Solo, in F:OTD, he is older and more battered. His history is also a bit more colorful, as he and the Doctor know each other and seem to have a cordial relationship of sorts.
I took the biggest liberties, I feel, with Professor Arturo. In Sliders, Arturo becomes one of the quartet of adventurers wandering from one parallel Earth to another. In F:OTD, that didn't happen. In his unrevealed backstory, our Arturo investigated the disappearance of Quinn Malory and discovered Malory's notes on sliding. When Malory never reappeared, Arturo took the notes and cobbled together his machine, trying to access other dimensions. Unfortunately, things went a bit pear-shaped for the professor.
Which leads us to the question of the breach. Was it a result of Arturo not quite understanding Malory's notes? Or, was it caused by some outside influence? The Other Universe, perhaps, from Fringe? Maybe the Kromaggs from Sliders were involved? Could it have been one of the Doctor's enemies? Perhaps it was the chaos-mage, Ethan Raynes, working under the auspices of the Black Guardian? Perhaps the Lovecraftian horrors from the Dungeon Dimensions were trying to break out?
The world may never know.