I realize how old this thread is but I'd like to provide my experience.
Close to 30 years ago I had a gunsmith open up a 375 H&H for an improved of my specifications. The increase in powder capacity is very similar to the 375 Weatherby and A.I.
My starting point for loading was the load data for these two cases. The resulting speeds surprised me.
Today's powders would have simply not encouraged me to improve the chamber of my rifle as they are very efficient.
Suffice it to say that without excessive pressure, the average speeds I obtained were as follows:
1) Nosler 260 grains partition, 3060 fps
2) 270 grans Fail Safe, 2950 fps
3) Nosler 300 grains partition, a hair over 2800 fps.
All loads chronographed with an Oehler 33 and an Oehler 35 P. Twice and with a friend's 35 P.
Barrel, 24 inches.
I do not wish to provide the loads as this is an individual rifle thing but if all you're getting are the speeds listed in the above posts, you aren't using the right powders. Quite frankly, I never expected to achieve these speeds, in fact nowhere near...Use slow burning powders with magnum primers. I used Winchester magnum primers.
I obtained some of my best results with Reloader powders. Reloader 19 for bullets up to 260 grains and Reloader 22 for heavier. Go safely, every single improved rifle is different but to only obtain 2650 fps with a 300 grains bullet after the improvement is a total waste. These are speeds easily achievable in any properly reloaded standard 375 H&H.
Before the improvement, I obtained 2750 fps with Speer 270 grains bullets, a regular boatail. (I didn't chronograph the 285 GS)
And also obtained 2700 with Sierra's 300 grains boat tail which is a bullet with minimal bearing surface. This definitely helped achieve higher speeds. These were hot loads but not excessively so. My point is that in the proper action, speeds achievable are not published. These were shot off the bench without a scope or muzzle brake. The rifle weighed 8.1/2pounds. The recoil was sharp but very tolerable and even with open sights, the groups were excellent. At the time I planned to use these bullets for North American hunting. I ended up using Partitions.
The recoil of the improved cartridge is sharper as it becomes a high intensity cartridge while the 375 H&H is looked at as an "efficient" cartridge. Don't expect the shove most people describe the H&H as providing but rather a sharp strong recoil approaching and at times exceeding 416 Remington recoil.
I had the barrel fitted with a muzzle brake and added a scope, both the added weight and brake made bench shooting quite tolerable for long sessions of 50-60 rounds. I don't hunt with the brake but I strongly suggest it for load development.
Each rifle is very different. Before making an improvement, see what you can get with modern powders and a regular 375 H&H. IMR 4350 (old Dupont) used to be the go to powder. It's still excellent for the standard case but there are also others that weren't in use 30 years ago that are excellent, however the added speed, in most improved rifles, are achieved with Reloader powders.
Barnes in their reloading manual #3 and Hornady in their current manual both offer load data for the 375 Weatherby, that particular information is a good starting point.
Load Data online also has A.I. loads. A balance between the milder loads of these sources are the way to go.