Ah well. I'd love to know how these razors got to America, but I'll be content with running into the brick wall of all the burned records of WWI soldiers. :)
On that note, I have turned up some more stuff.
First up is that mark I mentioned. Here's a picture of it on my razor:
Attachment 75573
Here's a picture of a similar mark on another razor:
Attachment 75572
I feel pretty confident saying those are the same (well except mine is WD1 and the other one is WD2).
That symbol in between the W and the D is a Broad Arrow or crowsfoot. It means that these razors were, unequivocally, military gear.
I've heard back from the Imperial War Museum, but I'm not sure how helpful they'll be. It looks like what they're offering is to set up an appointment to come look at their collection. If I had a passport and spare money for visiting London, I'd do this in a heartbeat. :)
So I've started doing deeper research on my own.
It looks like a lot of the service records were lost during bombings in WWII, but many do still exist. Plus there are secondary sources for things like the Royal Engineers and the Royal Field Artillery.
What I'm wondering now, though, is if those numbers by themselves can tell us anything. I notice on yours that the first two digits are separated a bit from the last three, and I can convince myself that mine has the same separation. So, I'm wondering if the first two digits tell us about division. If so, that suggests your razor is from the 26th division of the Royal Engineers.
But that doesn't seem to actually follow based on the pictures and resources I can find in the website with info about the 26th division. Also, there doesn't appear to have been a 77th division (though there was a 77th Battery associated with the RFA, it was part of the Indian army and stationed in Peshawar).
So, a bit more forward progress.
I'm still digging, and I'll be getting in touch with the guy from the Imperial War Museum to see if he can help.