I'm not sure what you are looking for in the way of a date, but as with many sets, the best we can do with this type of item is give you a range. As shown before, I have the following 4-0n-1 card also made by the Universal Toy & Novelty Co.
Seeing as these two items were found together (and sold by the same seller at the National) you can actually infer more from mine than yours. Speaker was first w/ Cleveland in 1916 and O'Neill was gone from Cleveland around 1923 so it is likely a 1916-23 or so card.
The other thread referencing these 4-on-1 cards still in the original Universal boxed set shows one w/ Cobb, Alexander, Johnson, and Bancroft on it. Unfortunately this doesn't help us much other than confirm it is 1915 or later (as Bancroft is on it). However, there was also a boxing 4-on-1 card that sheds a bit more light. Luis Firpo is apparently pictured there and while he was a famous boxer in South America, he really wasn't widely known in the US until 1920 or so & actually first fought in the US in 1922.
Also in the Universal boxed set that was found were some photos of an actress that looks like Bebe Daniels (very Betty Boop like in appearance). She was a very famous actress that began in pictures in 1919, but wasn't really a true star until 1920-21.
Given all the information that we have I would venture a guess that this issue of 4-on-1's were prduced by Universal Toy & Novelty between 1920-1923 or so (& most likely 1921-23). I have a really hard time thinking earlier than that, but it could be slightly after that, however, personally I don't think they were produced beyond 1925 or so.
So there you have my take on the cards. Likely 1920-23. Unfortunately, until more subjects come to light that range is the best we can do with the players pictured & the circumstantial evidence that we have.
-Rhett