You can go to to Grizzle tools and purchase the a PDF of the Production card that will give you the production date and original purchaser, look for the South Bend tab on their web site. It costs about 25.00 dollars. I would say it is a early1940s vintage South Bend 13 with the single tumbler gear box. If it has a plaque on the bed that states it was approved by the war board then it was produced during WW II. Also you can go on some of the machinist forums and download the catalogs from the time period and using the Model number get a idea of when it was produced. The production card from Grizzly is well worth the 25 dollars.
I just bought one and doing some research. The year 1950 started out with #1600 and the year 1955 started out with #5030. So your lathe #2637 was made between those dates. If they made the same number of 13” lathes per year yours was probably built in 1951. For 25 dollars Grizzly will provide you with a build sheet for your lathe so you know everything about your particular lathe including who bought it when new.
I have the same lathe, mine is from 1967, it looks like yours is probably from the late 50's. Grizzly can get you a scan of the original build card for $25.00 which will give the build date and a lot more.
Sweet! If you check out Dean over at the Ayr Cave Shop he did a very nice restore on his 13". I have a 13" as well, but a bit newer and with the double tumbler gear box.