I guess that, since we know it's Wimbledon footage from 1910 to 1919, we have to check the results pages and compare with every picture available of every player. Shouldn't be so hard, they weren't so many to compete in those times.
Based on pictures observation alone (which is certainly not enough but could be a start), I'd say this player serving underhand at 1:00 looks like Winifred Ramsey (Geraldine Beamish). So it'd be helpful to know if she was serving underhand in 1910 or after. In 1910 she lost first round to one M. Messom in Wimbledon (who could be the player before on the movie, since I have the impression we have two players facing each other judging from the same background - and same ballkid sit behind!).