Neat, huh?
I've done a little cursory poking around and haven't turned up anything.
And when I say "haven't turned up anything" I mean, there are zero Sheffield directory listings for ANYONE named Neat. Nor listings in the apprentice registry. So I think it's safe to say whoever that Neat was, they were not neat in Sheffield.
I did turn up one hit for a Charles Neat in Wales, but he was listed in 1841 as a Forgeman rather than a cutler, so it's very unlikely.
The name is kind of a challenge to search on for probably obvious reasons.