This is a fascinating video and I’ve been able to suss out some information on it, but not the exact date. Or dates, and you’ll see why after you watch it:
All right, first, it says “Chicago Cubs Game 1940’s” right on the title. I probably would have been able to figure that out from the uniforms. The title card says “1942.” The YouTube page says 1942 was the date found on the film stock, but of course 1942 film stock could easily have been used after that year.
Anyway, 1942 can’t be right, because at :35 you see a player wearing No.