Several removed comments have simply attempted to recommend "Blitzed" by Norman Ohler, a pop history of drug use in Nazi Germany. This violates the rules of the subreddit, as a book recommendation requires some discussion of the books quality to be allowed to stand, but not exactly why I am commenting. Given the related nature of the book to the topic, I feel it is quite relevant to link to a previous discussion of Ohler's work. It has been recently popular, but it has some major issues. /u/commiespaceinvader has addressed them here. Commie does briefly touch on addiction issues, but certainly is a topic that could still be expanded upon greatly.

That's an interesting question, one I haven't given much thought before which, in retrospect, seems odd to me, as (personal anecdote up front, sorry mods) my grandfather suffered from sleeping pill addiction his whole life following his Wehrmacht years. So I did a little research, this is what I found (it's not much - does anybody need a PhD thesis topic?), only about Germany, as that's my field of research.

First of all, you have to keep in mind that the number of Pervitin (Methamphetamine) users was heavily reduced towards the end of the war - first of all, the Reich had produced and shipped less and less of it (mainly after it became a controlled substance in 1941) and was, of course, able to produce fewer rations. Even if you discard the impossibility of purchasing addictive drugs, the end of the war meant a sure way of quitting anything and everything you were addicted to cold turkey - in May 1945, 11.5 million German soldiers were kept in PoW camps which made it next to impossible to still use drugs in any way.

When we look further ahead, into the first years of the Federal Republic, things get a bit more interesting: There were drug users, and it was public knowledge that there were, but they weren't frowned upon or seen as a crisis of society - mainly, because the numbers were lower than right before the war and because the German society still remembered the relative rise in drug addicts after the end of World War 1. There are even reliable numbers of drug users in Germany at hand: The United Nations Commission on Narcotic Drugs counted 5,228 Opiate addicts in 1954, sinking to 4,558 in 1959.^1 The commission named two important factors of the relative rise:

1. A quarter of those addicts were people employed in health care, mainly doctors and nurses. These people benefitted from the ease of supply, as Pervitin was still in use (it was a prescription drug in the Federal Republic until 1986)

2. Many addicts only received Opiates and Amphetamine during their treatment as disabled and wounded veterans, be it physical of psychological wounds to treat^2

According to Axel Groenemeyer, a german sociologist, the drug policies in Germany after 1949 were more progressive than they are today: Because the young republic had much more concrete problems to care about, and the number of users was so low, drug addiction wasn't seen as a social or political problem but a medical one - treatment of addicts was in the hands of doctors, not the police - of the 4,558 officially known opiate addicts in 1959, only 144 were sentenced by courts.

One last fun fact: Narcotics addiction in Germany in these times only meant classic opiates and amphetamines - which is a stark contrast to the USA - while cocaine and heroin saw a rise in the USA post-1945, they were irrelevant in Germany in the same way Cannabis was, they didn't even show up in the annual national drug reports.

So, to some things up: There were addicts, but most of them not through the use of drugs in the Wehrmacht. And most of those addicts were treated as patients, not criminals.

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References, unfortunately only in German (please don't read Ohler)

^1 Martin Schmid: Drogenhilfe in Deutschland. Entstehung und Entwicklung 1970-2000. Frankfurt 2000, p. 90.

^2 Axel Groenemeyer: Drogen, Drogenkonsum und Drogenabhängigkeit. In: Axel Groenemeyer (ed.): Handbuch soziale Probleme. Wiesbaden 1999, p. 444.