Not by any means an suthority on the subject - and I've seen displays of vintage .22 rounds at collector (cartridge and gun) shows. So there are those who know more than me.
I will give you what I have.
The green Winchester boxes predate my knowledge. Likely prior to the 1930s and possibly further back.
The green Win .22 short is likely older. The cover shows the numerals 15-10-8 suggesting 1915, October 8th.
The blue box of Win .22 short seems to be in my memory. In the possession of my late father or late brother. Probably purchased by them - again in my somewhat slippery memory - probably in the late 1930s or 1940s.
The Union Metallic Cartridge company was purchased by Remington (ammunition) in the early 1920s. The name was used for decades afterward, as I have centerfire cases head stamped "Rem-UMC", so that does not guarantee the box shown was made prior to 1923 or so. (Again, my memory.)
Winchester Wildcat and the Federal boxes I remember from purchasing in my youth. Late '60s, early '70s or so.
That is a somewhat shaky idea for you to start.