If it's thick wall, it will be a small-action BSA - and Trev is right, the barrel shank is too small for the round. The action will handle it though.
Barrel shank size is important here. A good number for a varmint round in a small-action BSA is the 5.6 x 50R, a bit like a lengthened .222 mag. It will get you close to 22-250 MV's.
And not to worry, it will warn you in time when pressure are getting too high - that's when the cases will be hard to extract. A Martini doesn't have the extracting power of a bolt action.
These small-frame actions are actually stronger than the large ones - better steel, thicker receiver walls.
Main limitations are barrel shank size, rimmed cartridge (not much of a limitation though) and cartridge length for chambering - especially with straight cartridges: it needs to be able to "make the bend"...
