I can only speak to my personal experience and I don't have any sort of advanced pilot's license: I just did the very first one - PP/ASEL. In that course even prior to instrument training I was taught to not fly by the seat of my pants.
The way my flight instructor explained it the human body is easily fooled and will quickly adapt to a new normal. So you had as an example a pilot getting used to a slight tilt and assuming this is normal, so they keep correcting for the tilt and worsening their situtation.
Even as a lowly PP/ASEL I'd still validate any reaction by reading the instruments before assuming a feeling of falling is accurate. Would that take thirteen seconds? Hopefully not, but it would take more than one.
Combined with the idea that airline pilots do less and less hand flying and also have many inputs in the cockpit trying to tell them what's happening and therefore may need a few seconds to integrate all that data and then react with a potentially rusty skill I'd feel comfortable saying a subsecond response seems unlikely.