The Williams date codes are here:

Williams component dating

Z means 1937, AW means 1956. I assume the numbers are the tooth counts of the rings.

Your Sturmey Archer trigger shifter is from the early 50's.

Here's what I'm thinking, about how the frame could have been made for a Sturmey Archer hub:



I definitely see one of the pulley wheels; I think I see a second one; I do not see the third at all. I see something that could be a cable stop, which would work with the system I'm describing, but it isn't normal; usually that cable stop (called a "fulcrum" in Sturmey Archer speak) is clamped on. The trigger shifter in your photo went on the handlebar. It's possible the original shifter was clamped directly to the top tube.

What this means, if I'm right, is your bike was originally set up as a 3 or 4 speed.