Generally speaking, French doesn't do double negation. A sentence is either negative or not. But it can be difficult to figure out, because the words involved can have multiple meanings, and because the language is changing.
In formal or literary French, there is only one word that conveys negation: ne (which is abbreviated to n' before a vowel sound). Other words such as pas, jamais, rien, personne, etc. are not negative in themselves, they only characterize what is being negated: a yes/no statement, a time period, a thing, a person, etc.
Je ne comprends rien. (I do not understand anything.)
Je ne comprends jamais rien. (I never understand anything. The sentence is negative because it contains ne, and the two words jamais and rien cause the negation to apply to both time and object.)
Je ne comprends plus jamais rien. (I never understand anything anymore.)
One complication is that when ne is paired with que, it takes a somewhat different meaning: it means “only”. In French, we see “only” as the negation of everything else, but most languages don't regard this as a negation.
Je ne vois que tes yeux. (I only see your eyes. In French, we think of it as “I do not see things other than your eyes”. There is a similar phrasing in English, but it's a lot less common than in French: “I see nothing but your eyes”.)
Adding a negation-characterization word doesn't flip the negation around.
Je ne vois jamais que tes yeux. (I only ever see your eyes.)
It can sometimes be hard to figure out whether a sentence contains the ne … que negation-like construction, or just happens to have a negative clause followed by another clause introduced by que.
Je ne comprends rien à ce que tu écris. (I understand nothing of what you are writing.)
Je ne comprends rien que ce que tu écris. (I understand nothing but what you are writing.)
Now that we have the necessary background, let's look at Druon's sentence.
Une tradition, ce n'est jamais qu'un progrès qui a réussi.
The verb est needs a complement (an attribut du sujet). The only possible way to parse the sentence is that this complement is the clause “un progrès qui a réussi”, and ne … que means “only”. As I explained above, jamais is not negative in itself, it only characterizes what is being negated.
Jamais is almost always about time, but in this case it isn't really about time. The expression “X n'est jamais que …” is an idiom that means roughly “X is only …, you shouldn't look for anything more complicated”. A more idiomatic translation is “X is nothing but …”. So the translation you found is correct, and it's mostly idiomatic except for “a progress” which doesn't really work in English (“progress” is not countable).
Just because this wasn't complicated enough, what I wrote above is not always true. The particle ne usually indicates negation, but sometimes it can mean … nothing. The “negation flavor” words jamais, rien, etc. can be negative on their own in some cases, for example in a short answer that doesn't contain a verb.
Est-ce que tu fumes ? — Jamais. (Do you smoke? — Never.)
Most importantly, in informal French, ne is omitted from negative sentences. So in informal French, the words pas, plus, jamais, rien, personne, etc. are really negative particles, and so is que when it means “only”.
Je comprends jamais rien. [informal] (I never understand anything.)
J'ai plus qu'une chose à dire. [informal] (I only have one thing left to say. By the way, in this sentence, the final S in plus is silent. To make things really confusing, you can say this sentence with the S in plus sounded, and then it means something completely different: “I have more than one thing to say”. The silent-S interpretation is the one that comes to mind when reading the sentence out of context, however.)