Does the order in which you finish Broken Age (Act I) affects the ending?

Solution 1:

It doesn't seem to: (skip to around 16:00)

I finished the same way you did, but this walkthrough shows what happens if Vella finishes first. Seems to be exactly the same, just in a different order.

I.e., each character's fainting is where they leave off, and then both rejoin for the final cutscene.