Here’s the fastest way I know to sew on a button by hand.
The key is to use four threads instead of two.
But how do you thread a needle with four threads without getting in a terrible tangle?
With four threads you can sew a two-hole button with three passes through the holes. A four-hole button can be sewn with 2 passes (for each pair of holes).
Some factories use six threads or even eight threads, to sew a button on in one motion (but I don’t approve!)
Here’s a 3 minute demo, from spool of thread to finished sewn button: