In my (limited) experience, it's pretty much instant.
A friend was buying a car earlier this year. Had arranged for the new insurance to start from 9am. We were in the post office at 9:30, and taxed it without any problems.
But I don't think if taxing it in the post office (which you'll have to do, as the V5 isn't in your name) that they actually check the insurance : it seems that if you're honest enough to be taxing a car, then you're assumed to be honest enough to insure it !
Do remember that bit : you will HAVE to tax it in the post office, using the V5 'green slip'. If the V5 isn't in your name, then it is not possible to tax it online.
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