^Are you discussing the actual time need for people to vote. Or the time need to calcualte those votes?
And where do you get the idea that you need two independant firms to audit it?
And what took days or weeks centuries ago, doesn't mean it has to take that long today.
It could be a simple case of a retina scan/thumb print/dna scan to verify a person and then pushing a button on a console. And when the polls close simply a matter of pushing a button to pull up the results.
True it could take days or weeks for some votes from the more remote ships and planets to reach Earth.
In an electronic form the results should be availble within hours, the actual process of voting could take days. Besides for more remote worlds they simply vote earlier. So if it takes 2 days for a subspace radio message to reach Earth, they vote 2 days befoe Earth.