Cogswell and Harrison’s were good working guns. Not best quality but affordable. Game department’s in Colonial Africa were equipped by Cogswell’s.
They were certainly not top end like Rigby’s, Hollands, Westley Richards etc. They were the sorts of guns used by farmers and game wardens, rather than those on Safari.
They are not really any different to any other rifle built on a Mauser action. Unless the stock is completely broken, I would think long and hard about replacing it. A gun is only new once, and a 375 H&H with a well used stock will have lots of memories and stories in there. It will most likely be straight grained and a little plain for modern tastes. But it will be strong.
My advice would be to leave it along, or to give it a gentle clean up. A light soap and water will get a lot of surface grime off. Wipe it dry, and then a few drops of a good gunstock oil every now and then will make it look many years younger.
Realistically for a new stock it will need be hand built. A decent blank of walnut will cost a few hundred quid, and then you add in the labour. In Zimbabwe, one of the ironwoods or mukwa might work. There was a gunshop in Harare that built and refurbished guns. But that was 20 plus years ago when I was last there.
I suspect it will have been stocked for open sights. If its scoped and you want a higher cheek piece, a simple lace on leather one is probably the best option. A permanent fix would be a piece of wood screwed on or let into the comb. Ditto for stock length.
Such adaptations are just part of keeping an old gun going. Treat it and use for what it is. If you want a modern 375, go and buy one.