I don't think you need to worry about much going all the way down to a 20" barrel in a .375 H&H, though mine is a 23" and it still hits all the right numbers as far as velocity vs. what the load data says it should be doing. There has been a lot of hoopla I've read in recent years about how short barreled guns don't burn all the powder in the barrel and spit it out at the end to produce the flash (some gun writers still cling stubbornly to this myth.) This is not true and has been proven false over and over since the 1940's but still the legend persists. You shouldn't worry about it. It is an urban legend. If you are getting too much flash, change your powder. Faster burning powders, as a rule, produce less flash because the pressure is lower at the muzzle.
All modern rifle powders are have burned all they are going to burn by the time the bullet reaches between the 6" and the 12" mark for the most part. If hasn't been burned up by that point, it isn't going to burn and will be spit out of the barrel unburned and scatter across the ground. Have you ever seen a snubby .44mag shooting heavy h-110 loads over snow? Thats what happens when not all the powder is burned up in the barrel. It looks like fleas everywhere.
Anyway, Big bores use their pressure much more efficiently than do long overbore cartridges like the Weatherby's and the RUM's. Velocity loss will be minimal until you get below 20". If you are having a flash problem, switch to powders that have a flash suppressant, or use a lower pressure loading. Flash is caused by the incomplete ("oxygen-poor") deflagration(burning) nature of most powders. the gases produced by the powders incomplete combustion are flammable (Nitrous Oxide, Carbon Monoxide, etc.) When they slam into the atmosphere at 2,500 degrees and mach 4,like they would from a short barrel, they naturally all go up once enough oxygen has been introduced unless they can be cooled rapidly by either a mechanical device (flash supressors break up the high temp shockwave proceeding the gas) or by insulating them much like a shielding gas (flash suppressant, different salts used to be used for this, I am not sure what they use now.) This is why muzzle flash appears to start several inches from the end of the muzzle. Black powder is still burning most of the time by the muzzle and you get the distinctive orange fireball emanating from the bore. Smokeless doesn't do this.
Anyway, back to your original question.... you shouldn't worry much about shortening barrels for cartridges that are not immensly overbore. Going really short has an effect on all cartridges so I wouldn't personally go below 20". Shortening a barrel also makes the barrel stiffer and decreases barrel whip, theoretically increasing accuracy but at the cost of some velocity (target shooters typically have longer barrels to increase velocity not accuracy.) This has diminishing returns as well but I have found that my 16" barreled guns usually shoot just as well and are less finicky than my 22-24" barreled guns.