The point of the stall warner is, you guessed it, to warn you. If you react properly to that warning, nothing should happen.
If there's a stiff wind then there may also be turbulence (vertical gusts) about from buildings etc. This may also cause the stall warner to bleep occasionally, as such vertical gusts temporarily change the AoA of the air.
So I wouldn't worry about that stall warner going off, unless the pilot did nothing to decrease his AoA (lower the nose, increase thrust).
I would be more worried about the "stage left" takeoff. A vintage taildragger, a pilot light on hours and a stiff crosswind might not have been the best of combinations.