Alas, the presence or absence of 'Sheffield' is not helpful. I think the earliest razors I've seen that used it were late 1820's, and then it was used inconsistently ever since. Rather than a legally mandated mark (like the much later country-of-origin stamps), it was simply to reflect pride in the place it was made and to capitalize on Sheffield's reputation for excellent cutlery. In short, a marketing slogan not hugely different than Silver Steel or Universally Approved.
The Cut A Way razor, I agree, definitely looks like a Sheffield razor -- especially that cartouche around the name.
I poked around a little outside the usual sources (trade directories, Tweedale's directory, etc), and found another one sold in an auction.
Attachment 315032
Attachment 315033
Attachment 315034
Pretty clearly the same maker, but again with no clues as to who it actually
was. It does at least contribute the data point that they are likely to have been in Cornwall (though the blade could still easily have been manufactured in Sheffield).
I'd say that's a rare, and quite interesting one! I'll keep doing what I can to find the manufacturer, but currently I don't have quite enough to go by (there were a lot of cutlers in Cornwall).