What happens in that situation is the airspeed drops, and with it lift (which is proportional to V-squared), and the airplane then drops without changing pitch attitude. That increases AoA, possibly to the stall point. That's why the first thing to do in a LLWS airspeed loss is add power to increase excess power and prevent the airplane from dropping before the airspeed recovers. The good news in light singles is their sectional density is so low and engine response so fast, about all you see is a momentary blip in airspeed before recovery. It's the big heavy jets with high sectional density and long spool-up times which have the big problems.
A nice explanation.
As Ron says, big jets with long spool times are in trouble in those conditions. Recovery is pretty quick in a light piston single but not instantaneous.
No stopwatch handy, obviously, but Doug did the correct recovery (throttle up, and definitely don't pull up! -- he didn't have to push the nose down, but we had a little altitude left to trade if needed), I'd guess I counted about 2-3 seconds at 45 knots indicated and a blip of the stall warning. We still had some margin, but not much.
(Stall warning comes in well before the stall in a Skyhawk if you're not pitching heavily, and the difference between calibrated and indicated airspeed increases in a Cessna at low speeds, both indicated that we were still flying... plus the floor wasn't dropping out by "calibrated butt feel".)
Once it recovers you have to be ready to get the throttle back out or it's going to accelerate above your target speed (approach in this case) with all that extra power in, and it'll do that as quickly as you're used to seeing under normal conditions.
Doug was originally flying it about 65 and slightly slowing, when I said "fifty-five, FOURTY-FIVE" about as fast as you can say that phrase at a non-panic'ed speed, to give a feel for how quick it occurs.
I'd been chanting sixty-five for about three repeats, after he was stabilized -- and was about to say "sixty" when I said the other two, back to back.
At the 55 call he shoved power in. The power stopped the trend (movement) downward of the ASI at 45 and within a couple of seconds it was trending the other direction and he then put the power back where it was, I'm sure just by ear and feel. (He didn't have time to look inside, I called numbers, he landed in the crosswind.)
He later said he was on a go-around hair trigger and the landing was one of those where, "I wasn't going to land... until I did."
Without someone calling out airspeeds so he could have stayed visual (for the crosswind), he said he would have gone around and asked for the crosswind runway.