How did "at once" get to be used to mean "immediately"?

The sixth and final definition of at once in the OED means "immediately, straightway" and is first documented in William Tyndale's Expos. & Notes way back in 1531:

The apostles were clear-eyed, and espied antichrist at once.

It comes from a meaning of "at one time" and other definitions of at once are of things happening at at the same time; or in one heap or body together; or at the same time or simultaneously.

So the "immediately" sense can be similarly thought of as things happening all at the same time, or right now.

Join the army at once & help to stop an air raid

Or:

Join the army right now & help to stop an air raid

You read the poster and then join the army all at the same time, or immediately.

Since the literal usage means two things happening concurrently, according to Wiktionary, I'd guess that the idiomatic usage assumes that right now is one of the concurrent events.