There is no name and there is a picture of one in the old Bannerman catalog which only lists it as an antique cannon.
From what time frame is the catalog? That might narrow things down a bit. The rate of twist (you can measure it with a rod and a ruler) might give some further evidence by suggesting what sort of projectile the thing was meant to shoot.
Some things about this gun show up to my eye as possible clues.
- The lockwork is as simple and crude as can be, and outside the gun where it can be damaged easily. It was, though, very easy to fabricate.
- The receiver appears to be essentially a billet with minimal shaping, not much there of form following function--unless the main goal was quick cheap fabrication.
- The muzzle is belled. Curious. Was the barrel recycled from an older weapon, or made that way out of a lingering stylistic habit or preference? If the latter, it may be a clue to the weapon's place of origin.
- The breech screw is continuous, not the more convenient (and higher tech) interrupted thread type.
- Screw breeches are more often seen on guns wherein the barrel is the main body of the weapon, rather than on tip-up guns on (improvised?) receiver blocks. If the receiver serves any purpose here, other than simply holding the lockwork, it is to mitigate recoil: It's a big chunk of metal.
All that suggests a low tech, low cost fabrication effort somewhere, for some purpose. None of it points to professional work, unless the professionals were constrained by serious problems of time and tools. Perhaps it was a quickie job in some armory somewhere? Or even a field improvisation in some shop appropriated for the purpose, a carriage shop or a boiler factory perhaps, where minimal trouble in building the gun was an essential factor?
That is my long-winded way of saying "I don't know." The points mentioned may, though, spark someone else's thinking. Quite often, THR solves these problems because somewhere, sometime, someone has seen something of the kind before, or heard of something similar.
Or, one last hazarded guess? It might, conceivably, be something Bannerman's threw together just to sell.
Some of their rifles had Frankensteinian forebears.