I have just been through this same exercise (again) of picking a scope for the third .375 H&H in my gun safe. My first .375 wears an older Bausch and Lomb Balvar 1.5-6x36 scope. Bushnell bought the B&L line of scopes, so that the same basic scope today is a Bushnell Elite 4200 1.5-6x36. The bigger objective lens of these scopes gives better low light visibility than the 20 mm objectives on most 1-5 scopes. I used my first .375 with the B&L scope to shoot several animals last month in Zimbabwe. It is a wonderful combination, even in the very low light conditions that existed when I shot my cape buffalo on the next to the last evening of the hunt.
The main drawback to the B&L 1.5-6 scope is the same as you will find on most of the scopes that have been mentioned by others in this thread, including the expensive European scopes (except Schmidt & Bender). Most of the scopes start at around 3 1/3" of eye relief at high power and end up at 4+" of eye relief at low power. This length of eye relief can result in the scope being slammed into your forehead and blood running into your eyes under the wrong conditions. That is exactly what happened when I used an identical B&L 1.5-6 scope on my .416 Rem Mag from a sitting position and later from a rest where I was bent over and leaning forward. The main difference to notice here is that the .416 has MUCH more recoil than a .375, despite what the gun writers may occasionally say. I love the .416 (I have two!), but both of them now wear the Leupold LPS 1.5-6x40 30mm tube scopes that have a more consistent and longer eye relief, between 4" and 4 1/2" depending on the power setting. My second .375 also wears an LPS 1.5-6. The LPS scopes solved the scope cut problems on the .416. Unfortunately, Leupold discontinued the LPS in 1.5-6. It was the best of the LPS line because it filled a void left by every other scope in this arena. In addition, the LPS scope has every desirable feature you would want in a dangerous game scope, since weight is the only potential drawback to an LPS and that is not an issue in the dangerous game category. If you can find the LPS 1.5-6, BUY IT! If you can't find one, don't worry, the .375 is fine with the normal eye relief unless you are shooting at high power from a prone or uphill sitting position. Try to use low power where possible to extend the eye relief.
I have had great results with the B&L Balvar 1.5-6 and decided to scope my latest (third) .375 with the newer Bushnell Elite 4200 1.5-6 with the Firefly reticle. I like it better than the Leupold VX-III in 1.75-6x32 for various reasons. The glowing Firefly reticle is nice if you are hunting lion or leopard in VERY low light conditions. Just shine your Surefire flashlight on the reticle for a few seconds to light it up for as long as you will need at any one sitting. There are no batteries to deal with or wires and lights inside the scope. The illumination is simple and foolproof. The scope also has more room for ring mounting than the original Leupold 1.75-6, although in later years the front tube on the Leupold was lengthened. The bigger objective on the Bushnell Elite is larger and gathers more light. The lower power setting on the Bushnell Elite is nice but not a big deal. The optics are very similar. Just so you know, most of my scopes are Leupolds, with my favorite for most big game hunting being the VX-III 2.5-8x36 -- a wonderful, compact, light scope. I use Swarovski binoculars and spotting scope and Leica binoculars and rangefinders (including the newest 10x50 Geovids) exclusively for my other optics, so don't think I avoid the European brands or am adverse to spending the money for the best optics for the job. I just don't think there is any great advantage to the European rifle scopes for my uses. I do believe Swarovski PH scopes have the nicest optics of any scope made, but this is an aiming device, not a viewing device. If I want to admire or judge the game, I will do it through my Swarovski and Leica binoculars or my Swarovski STS-80 HD spotting scope. If you think a Swarovski, or its less expensive cousin, the Kahles, is what would work best for you, spend the money and enjoy the incredible optics like Big Bore did. He has a gorgeous custom rifle that should wear incredible glass like the Kahles. I do not like first plane reticles, but they are supposed to be better from an internal engineering standpoint. Like I said, the fact that most of the European scopes do not have a lot of eye relief is not a big deal most of the time in a .375. It is a potential problem on a .416 Rem Mag or a .458 Lott or any of the other big recoiling rifles like the big bore Weatherbys. What I like about the discontinued LPS 1.5-6 is that you get the premium optics, second plane reticle, and long eye relief -- the best of all worlds in one scope. I do agree that the optical quality is still not as good as the Swarovski PH and some of the other European scopes, but it is definitely close enough. The Schmidt & Bender 1.5-6x42 is a good alternative in a super-premium scope because it has great optics, a 30 mm tube, perfect power range, good light gathering and excellent eye relief. I do not think the optics are as good as the Swarovski, but some people think they are better. I still don't like the first plane reticle and would prefer not to spend 4 times as much as for premium American optics and double the cost of even the LPS, but the S&B is a good choice in my opinion.
I usually have a second scope that I take with me for my hunts with the bigger calibers. I am probably going to buy another scope as a spare for the newest .375 which I will sight in and have mounted in quick detachable mounts. I like the Leupold QD mounts, but the Talleys are as good or better. Either one is perfect. My choices for the new scope have narrowed to just a few. If you want a Leupold, I would encourage you to look at the Leupold "European" (I will abbreviate them as LE) 30 mm scopes. They have much better eye relief than the VX-III 1.75-6. The eye relief range for the LE's is about 3 3/4" to 4 1/2". That is plenty for any shooting position on a .375 H&H and is adequate for almost any usable position on the real big bores that might wear a scope. There is a 1.25-4x20 LE and a 2-7x33 LE. The lower power scope would be good for a true big bore, but I would prefer the 2-7 LE as a better compromise for the .375. It is rare that you would ever need anything less than 2 power on a .375, although I would like a 1.25 or 1.5 if everything else were equal. It is more likely that you would want 6 or 7 power for longer plains game shots with a .375 than feeling like you needed less than 2 power. Also, the larger objective and greater light gathering ability is important for the type of hunting that a .375 does very well. If you buy a .458 Lott to hunt elephant and buffalo in thick jess, go with the low power scope!
Another scope to consider is the Burris Signature Safari 1.75-5x32. It has great eye relief, the "Posi-lock" reticle retention system, and good optics, all at a very reasonable price. I have talked to a number of knowledgeable people who think it is the best dangerous game rifle scope on the market. I will probably buy one for my extra scope on the .375. The reticle adjustments are only in 1/2 MOA, but that is no detraction on a large-medium or large caliber rifle. Just as the .375 is a nearly perfect compromise rifle, the Burris 1.75-5 is a nearly perfect compromise scope for the .375. It would be nice to have both lower and higher magnification and to have a 30 mm tube with Swarovski quaility optics and a Firefly-like illuminated reticle, but the Burris has everything you need and more at a very good price point. If you were to buy both the Burris Signature Safari scope and the Bushnell Elite scope, you would still pay about half of the cost of a single Kahles scope and would have a spare in case something went wrong with one of the scopes. Burris and Bushnell each have very good warranty coverage, although you would probably never need it. I still recommend the LPS 1.5-6, but since you can't get it, one or more of these others will do the trick. The good thing about the .375 is that its relatively mild recoil does not present a huge problem for any of the scopes that have been discussed in this thread.
I thought you might be able to use the thoughts of someone who has been in your shoes on three occasions with three .375's as well as several actual big bore rifles. I sold my .458 because it did not have the versatility of the .416's, and since I needed to always have a "do-everything" rifle in my hands while I am hunting in Africa, I decided to get rid of the more specialized caliber. Now I always take a .338 and either a .375 or a .416 to Africa with me. If you just take one rifle on safari, make it the .375. The .338 is my favorite caliber and perfect for anything and everything smaller than cape buffalo, although it will kill anything with the right bullets and good shot placement. Take some solids for the .338 just in case you need to stop an aggressive elephant, buffalo or hippo. A tracker was tusked by a charging elephant days before I arrived in camp because the client and PH had left the big rifle in the vehicle while hunting plains game when the unprovoked charge occurred. A 180 gr softpoint from a .30-06 will not penetrate the skull of an elephant far enough to reach the brain, as the death of the tracker proved once again. The .416 kicks harder than the .458, but sure shoots a lot flatter, and it will definitely put an elephant on the ground if the shot is good.