If an Adversary/Event/Scenario Action causes an Invader Action, does "Pour Time Sideways" double both the the original Action and then twice double the Invader Action?

If an Adversary/Event/Scenario Action causes an Invader Action, does "Pour Time Sideways" double both the the original Action and then twice double the Invader Action?

No, "Pour Time Sideways" doesn't double anything that's already doubled.
For example, if time is poured onto a board twice, then "On each board: Choose a land with Cities. Build there." (the "Provincial Seat" Event) makes players choose a land on that board three times, but only Builds once in each chosen land for a total for three Build Actions on that board (rather than causing nine Build Actions by choosing three times and Building three times per choice).
Formally, when "Pour Time Sideways" changes how many times an action happens, it doesn’t apply to any secondary (triggered/nested) actions caused, directly or indirectly by that action (i.e., any descendant in the action tree).