There really isn't any question whose care you want to come under, and the answer is plainly the Western Allies, for one simple reason. They had penicillin, and everyone else didn't.

Although it was discovered in 1928, turning the mold that Alexander Fleming discovered into something practical for medical application took a lot of time. It was a decade before scientists working at Oxford under Howard Florey even understood how to move forward with it, and the was had started by the time animal testing began in 1940, essentially cutting Germany off from any access to the developments. In early 1941, enough penicillin existed to treat a total of six patients, and not all of them survived those first clinical tests (although the tests were still considered a major success). Further development was mostly done in the United States due to the wartime situation in the UK (which essentially handed the forefront of research to the US), and it was there that the techniques for mass production were perfected (I won't pretend to understand the sciency stuff, but it had to do with "submerged culture fermentation").

It wasn't until 1943 that production was able to get into full swing, but it certainly did. Multiple companies in both the UK and USA were all turning out millions of units a month. Over 35 groups would be producing the drug by the end of the war. And while it was quite pricey, especially at the start, the payoff was absolutely worth it. The US government alone spent 14 million dollars on it, but at the end of the war production was in 176.9 billion units per month, with a 98 percent reduction in the cost of production (although it should be noted that one unit doesn't equal one dose).^1 ^3

The Germans on the other hand didn't enjoy the same success. They had access to the laboratory work of the British scientists only up to 1939, which cut them off from the breakthroughs in practical application and production. The British, in fact, took great care to prevent the secrets of penicillin from being stolen by German intelligence. Post-war analysis of German scientific research revealed that at best, small scale production had been attempted, but the Germans had not been able to put the submerged cultures necessary for mass production into effect. What limited amount they had made was in too small a quantity to even provide to the military. Lacking penicillin, the Germans relied on sulphonamides (sulfa powder, for you war movie aficionados) as their primary anti-bacterial, which was no where near as effective in treatment. Captured stocks of Allied medical supplies were greatly prized (although the medicine would lose potency quickly).^2

Now obviously, penicillin didn't save every single person who got wounded, and of course it only was available in any meaningful quantity from late 1943 onwards, but the effect simply can't be ignored. Its obviously a fluff piece advertisement, but the numbers presented in this chart still go a long way to demonstrate how much had changed in only a generation^4 Exact numbers are not very easy to find, although the uncited "12 to 15 percent" in the Wiki article seems repeated often without sourcing. The closest I could track down to a hard number was that "the healing rates of bone infections and compound fractures of wounded soldiers in WWI (25%) to those with similar wounds WWII, after penicillin became available (95%)". The author seems to be a professor of biology, but his website doesn't exactly scream "trustworthy!".

Aside from practical penicillin, other notable breakthroughs in medicine that were mostly just availale to the Allies included blood plasma transfusion, the insecticide DDT, and a notable number of vaccines (about 1/3 of all diseases which became vaccinated against during the 20th century) - botulinum toxoid and Japanese encephalitis, as well as improvements or replacements to the existing ones for yellow fever, cholera, smallpox, typhus and tetanus.^5 ^6

(One thing that comes to mind where the Allies had a disadvantage was quinnine. The Japanese controlled almost the entire supply for it, and atabrine, the substitute, was even more disgusting. Many soldiers got ill from taking it.)^7

Now, what all of this is to say is that, the Western Allies had medical care that was much, much better than the Germans, let alone the Japanese. Penicillin is the main reason for this, and its importance in saving tens of thousands of lives can't be understated.

EDIT: Doing some more digging today I was able to find a few more statistics concerning the effectiveness of antibiotics. The most striking, probably is that Acute respiratory diseases killed 50,000 American soldiers in World War I, but merely 1,265 in World War II, but this is credited to both penicillin and the less effective sulfa drugs.^8

Even more striking is the difference between penicillin and sulfa in regards to gangrene. American wounded developed 1.5 cases per thousand, and those who did develop it had twice the survival rate as in World War I. Germans, using only sulfas, developed 20 to 30 cases of gangrene per thousand! That is an order of magnitude higher!!^8

Sources

^1 The Search for Synthetic Penicillin during World War II - John Patrick Swann - The British Journal for the History of Science, Vol. 16, No. 2 (Jul., 1983), pp. 154-190

^2 Allied intelligence reports on wartime German penicillin research and production - Gilbert Shama, Jonathan Reinarz -Historical Studies in the Physical and Biological Sciences, Vol. 32, No. 2 (2002), pp. 347-367

^3 The British Success with Penicillin - Jonathan Liebenau - Social Studies of Science, Vol. 17, No. 1 (Feb., 1987), pp. 69-86

^4 The Penicillin Mystique and the Popular Press (1935-1950) - David P. Adams - Pharmacy in History, Vol. 26, No. 3 (1984), pp. 134-142

^5 Vaccine Innovation: Lessons from World War II - Kendall Hoyt - Journal of Public Health Policy, Vol. 27, No. 1 (2006), pp. 38-57

^6 Powdered Blood Plasma May Replace Blood Banks - The Science News-Letter, Vol. 38, No. 16, Fall Book Number (Oct. 19, 1940), p. 243

^7 An Army at Dawn - Rick Atkinson

^8 Conniff, R. (2013, 07). PENICILLIN WONDER DRUG OF WWII. Military History, 30, 38-43.