So last night I made a quick chart summarizing UHV timeline for all civs. I went by in-game descriptions so it's possible I was misled by the wording, but otherwise I went with three categories for UHVs:
-IN, which have to be completed at a specific date,
-BY, which can be completed earlier,
-OTH, other UHVs that don't have a specific limit (and are the most problematic since how early you can complete them depends on your skill, and there may be a hard limit).
Assuming unlimited civ switchs are allowed (which can be fixed in WorldBuilder's options and should be the default in later releases if I understood Leoreth correctly), then I think it's in theory possible to complete the UHV for 6 civs in the same game:
* Harappa:
-Spawn 3000 BC,
-BY 800 BC.
* Rome:
-Spawn 760 BC,
-IN 320 AD,
-OTH: "Be the first to discover Architecture, Politics, Scholarship, Machinery and Civil Service".
* Vikings:
-Spawn 545 AD,
-IN 1050 AD,
-BY: "Acquire 3000 gold by pillaging, conquering cities and sinking ships by 1500 AD".
* Mongolia:
-Spawn 1190 AD,
-IN 1300 AD,
-BY: "Control 12% of world territory by 1500 AD".
* Dutch:
-Spawn 1580 AD,
-BY: "Acquire seven Spice resources by 1775 AD".
* Then any of the latest civs (America, Argentina, Mexico, Colombia, Brazil or Canada).
Most of these transitions are unproblematic since the UHV ends before the next civ starts. The exceptions are Rome-to-Vikings and Vikings-to-Mongolia. The challenge is getting the Roman techs before 545 AD and the Viking gold before 1190 AD. The former sounds possible with smart tech progression, the latter should be made easier if you prepare as Rome to leave a lot of lightly defended cities and ships.