I'm not talking about scytale, but encryption like RSA, DES, etc.
How exactly did civil cryptography evolve after World War II?
I won't answer your question up to every detail, as I would have to write a book to answer the pretty broad question to full length. But I'll give you some hints as it would be wrong to let you think that non-military cryptography has appeared in 50's and 60's only thanks to leaks from the NSA!
Before the 30's…
One of the earliest descriptions of encryption by substitution appears in the Kama-sutra, a text written in the 4th century AD by the Brahmin scholar Vatsyayana, but based on manuscripts dating back to the 4th century BC. The Kama-sutra recommends that women should study 64 arts, including "mlecchita-vikalpa" — the art of secret writing*, advocated in order to help women conceal the details of their liaisons.
All in all, "non-military cryptography" has been known to civilians for thousands of years!
Where "Modern" crypto came from…
As for "modern-day" encryption, I'll only add that in the 30's and 40's, things like the Enigma cryptographic machine series were available to the general public, but they were too expensive for the average working man. Even after WW2, most military cryptography still was not all too different from the substitution ciphers known from the past, which were - theoretically - also available to every civilian. As an example, you can look at the work of Claude Shannon, who worked for Bell Labs and not the military.
Shannon worked at Bell Labs for several years. During his time there, he produced an article entitled “A mathematical theory of cryptography”. This article was written in 1945 and eventually was published in the Bell System Technical Journal in 1949. This is regarded to be the start of modern crypto… which is why Claude E. Shannon is considered by many to be the father of mathematical cryptography.
The (r)evolution of civilian modern crypto…
The cryptographic techniques and algorithms evolved in the 70's as computational power became available… and even "working class" civilians could afford computers. What leaked in those decades was nothing more than a hand full of different military cryptographic algorithms; but it's not as if there weren't any civilian cryptographic algorithms available at that time. As an example, you can look at the DES cipher you mention. DES was created by a research group at IBM, not the military.
Re-reading your question, I guess you don't know (or forgot) that it was not until the 1970s, when the US government started treating cryptographic algorithms and software as munitions and interfering with university research in cryptography, that cryptography was actually pretty "non-military"; especially from a research and development point of view. My answer to "Why are cryptography algorithms not exported to certain countries?" dives a bit more into this.
But generally it's a known fact that there has always been ample civil research and development of cryptographic algorithms and techniques and most military cryptography is based on and/or derived from such civilian research and development. An example of a modern crypto invention in the civilian area actually resulted in Bernstein v. US Department of Justice (which made it famous in the first place). Bernstein's "Snuffle" was not and never has been created, influenced, manipulated, or leaked by the NSA or the US government… nevertheless, if got hit hard by US legislation. So yes, there was influence by governments and their military… but they were not the ones who gave birth to modern civilian crypto. Shannon did that in '49 with his publication, and - just like thousands of years before - it's mostly civilians (especially academics and the likes) that invented and still invent modern crypto algorithms.
If you need more details, you should probably check Wikipedia's "History of cryptography" with a section called "Modern Cryptography". If that's not enough, maybe it's time to buy a book about crypto history. Wikipedia has a nice listing or related books at "Books on cryptography ~ History of cryptography".
Last but not least, I would like to add that it should be clear that the NSA (since you explicitly mentioned them in the question) had no chance to influence crypto before November 4, 1952… as they were founded on that date.
The history of RSA is that somebody working for British intel created something like it a few years earlier than the MIT guys who published publicly. Unfortunately for him, his work was so good that it was classified and kept secret until much later. That's the opposite of a leak, so your answer is no. Historically, cryptography was far less sophisticated than anything today but far less in use because it had to be manually implemented by some human being. That human had to be both pretty smart/educated and you had to trust them, or do it yourself if you could, so it was mostly (with some exceptions) for military or royal/government/diplomatic purposes centuries ago for that reason. It's more in use nowadays because computers just do it for you automatically, and anyone can set up an SSL connection by just clicking on the link to go an HTTPS protected page.
From the 40s through 60s, civilian encryption simply wasn't a pressing need for the world of its day. Digital communications networks didn't exist, so manual cryptosystems like book codes sufficed for most commercial operations. International transactions were carried by voice over long distance lines, not by machines.
Hebern and other rotor machines were commercially available, although they were primarily used for diplomatic missions. Shannon published seminal work in his 1949 paper, "Communication Theory of Secrecy Systems", in which he proved that the one-time pad was the only theoretically secure cryptosystem. As more digital communications developed in the 60s, as the rise of multinational corporations arrived, and as information became ever more valuable, the need for security grew.
Shannon's discoveries also included theories of compression and hashing systems, both used in cryptographic systems today.
Keep in mind that in the 1960s, a computer would fill a room, and one would serve a large company. Until the arrival of integrated circuits, computing power was not something available to individual people. Civilian cryptographic systems of the day remained primarily manual. And computer security was something achieved by keeping the door of the computer room locked.
But networking of computers wasn't far away. Remote terminals were spreading, and people began doing valuable work away from the office. There still wasn't much thought to security - simple passwords that prevented access were the rule of the day.
In the 1960s and 1970s, Horst Feistel laid the groundwork for block cypher systems, culminating in the creation of Lucifer for IBM. As e-sushi mentions, the NSA assisted by strengthening it against an attack that remained unknown in the civilian cryptographic world. Lucifer was adopted as the DES. (Ironically, until differential cryptanalysis was independently discovered in the 1990s, the NSA was accused of intentionally weakening DES with their tweaks.)
In 1976, Diffie and Hellman published their paper on asymmetric key exchanging, and the civilian cryptographic world changed again.
Other people have contributed heavily, of course, but those were the key players, and much of what we know today was built on those foundations.
Somewhere between the scytale and RSA there is the Enigma machine: It was commercially available since 1923; its military application came after the commercial one.