The timeline, as written and posted by LettuceBacon&Tomato:
2009: 3rd roleplay finishes. After learning about the bridge, the government begins planning and constructing a worldwide network of them, publically crediting Dr. Zanasiu, Dr. Kerzach, and Dr. Shelton for its invention since they'd rather not credit a deranged terrorist leader. (This is one of the reasons Pomson seeks revenge.) The bridges are unable to leave the Earth's surface or go backwards in time (in episode 6 Kerzach claims they did that because they deemed humanity unready, but in Pavlov's Checkmate remember Shelton said he didn't know how to replicate the time aspect of the original bridge. )
2022: Shelton dies of his disease at 53.
Between 2022 and 2023: Drs. Zanasiu and Kerzach learn of Hans' death and how it exactly matched Shelton's (its implied that other characters from the old team died similarly, further tipping them off.) Not wanting to go public because they know the bridge network (an amazing technological achievement) would be shut down, they decide to solve the problem themselves. Dr. Zanasiu will be put in time limbo (remember at the end of PavlovsCheckmate Moby offers James one free jump through the bridge for his help stopping Ricky.) And once the cure is discovered, he'll be cured and returned to the 'present'. (Also required to be invented is a lock that keeps the bridge open, so he can be returned.)
2033: their children, previously completely in the dark as to all this, learn enough to interrupt Dr. Zanasiu's time stream. Luckily Malte, who witnesses Dr. Zanasiu's return, pieces together the rest of the story and begins inventing the bridge lock. Dr. Zanasiu's antibodies provide the cure needed to cure Shelton's disease before it becomes an epidemic.
(2039: Dr. Zanasiu, Sharon, and Erik watch a film at a movie theater reunion. Junior is not present as he is stuck in the time stream right now.)
2053: Malte, who has been hired as a government scientist for his acquired knowledge in bridge mechanics, completes the bridge locking mechanism, meaning they can finally bring Dr. Zanasiu out. So they do, and then return him to 2033, and the only downside is that Junior and Florence are stuck in the future.