Hi. I just bought a CWS bike in San Jose, CA today at the flea market. I have found you and the guy that runs the Oldbike.eu online museum I believe he is in England and he only has two CWS bikes and talks a lot about fun it was researching the bikes so it does not seem even he knew much about them. I am not a collector but this bike just made me buy it. The cooperative story is kind of cool too. I am charging my phone right now but will post some pics when it's charged up. It looks very much like the one you identify as a 1951 though I am thinking you are way off. I am thinking they are late 1930s. The ONLY difference I can see between the one you call a "1951" and one I purchased is the sprocket pattern. The stamped number on mine is in the same location as yours and it is 7815. I am thinking this indicates Model 7 which came out in 1936.. CWS also produced Federal and Hurricane brand (the girls bike above).
Also, what do you make of the frame construction differences between the bolted rear of the "1951" and the brazed/welded rear of the "1948"
This link is an advertisement for the Model 7 from June 1936. You can see it is the same exact model, including 3 speed, as the one you're calling 1951. I think you are going to be happy it didnt sell. I think you might be right on this one being a 1948 but I think the other one is earlier.
https://www.gracesguide.co.uk/images/5/5e/Im19360603Cy-CWS.jpg
Here is a 1951 Raleigh catalog where you can see the brakes have already changed.
https://oldbike.wordpress.com/1951-raleigh-catalogue/
Here you have one that looks similar to your "1951" which I think is 1930s Model 7 and it was a barn find in England and the guy says his dad has been in bicycle industry for 30 years and never seen one before.
https://www.gumtree.com/p/bicycles/cws-cycle-works-vintage-antique-bike-bicycle-like-bsa/1233950659
Here is a link to one that rolled through an Illinois bike shop the OP dates to late 40s-early50s. He may be incorrect but two big changes are the metal head badge and the brakes. Gents version of the girls CWS Hurricane bike posted above by Fat Tire Trader
https://groups.google.com/forum/#!topic/gentlemancyclist/PsbWpv9MUF0
Here is the oldbike.eu guys bikes
A 1939 Tradesman Delivery Cycle
https://oldbike.wordpress.com/1939-cws-co-op-delivery-cycle/
More info and literature/ads for CWS
https://www.gracesguide.co.uk/C._W._S._Cycles
https://www.gracesguide.co.uk/Co-operative_Wholesale_Society_(CWS)
In case you are interested, here is some info on the CWS motorcycles.
https://vintagemotorcyclecwsfederal1922.wordpress.com/