Gibberish spoken by English-speakers, despite being intended to sound unruly, clearly has rules. I'm not a linguist, and can't find a great deal of research into this, but here's a paper linking the "funniness" of a word to the extent that it stands out from normal English, which seems to conform with experience. Experience would also suggest that there are at least two forms of English gibberish, one that attempts to be as flamboyant as possible, such as in Dr Seuss, and another which attempts to mimic sillier-sounding, archaic parts of English, as in Jabberwocky. The former seems to use attention-grabbing techniques, especially prosody. Obviously, one would expect gibberish in other languages to stick to phonemes used in those languages, but I'm wondering about more profound differences. One would expect other forms of gibberish to perhaps mimic their actual corresponding languages, or to use different elements, less likely to appear in the corresponding language. For example, English gibberish uses repetition heavily, we already have the two examples of exact repetition alone, "Jubjub" and "Tumtum" in Jabberwocky. We essentially never repeat the same word in English. However, in other languages such as Indonesian, Malaysian and some Australian Aboriginal languages, repetition is part of standard grammar used for emphasis or to indicate plurals. One might thus expect gibberish corresponding to these languages to have less of an emphasis on repetition.
First you have to define what you mean by "Gibberish". In the modern era, there has been a huge expansion of varieties of gibberish in the form of con-langs such as Klingon, Na'vi or Dothraki. On the one hand, you might want to exclude "deliberately-constructed languages" from the scope of investigation, on the other hand that could define "gibberish" out of existence. It's also not obvious how you would treat language games (Pig Latin, Ob, Røverspråk – one of them, a transformation of English – is actually called "Gibberish"). Then finally, there is a common process of mock-language creation, whereby a person makes stuff up that they think sounds like a specific language (c.f. Mock Swedish spoken by "The Swedish Chef"; various mock languages in old b-grade movies where they didn't bother to get an actual speaker of Comanche).
In deciding what kinds of things you want to include in your study, you should start by filtering out the most-contrived language transformations, thus eliminating con-langs and also the careful creations of an author, such as Lewis Carroll, who put a lot of thought into Jabberwocky. The reason for eliminating the most-deliberate constructions is that you won't learn anything about the relationship between a particular natural language like English or Swahili and the rules of an intentionally-created code, you will learn about a particular author. (One can also independently study why an individually-created language has the properties that it has, by studying David Peterson or Marc Okrand).
Language games, which are conventionalized filters applied to natural languages, are very wide-spread and also well-studied to the point that we can say what are the rules of Jeringonza (actually a huge collection of similar dialects) or Pig Latin (again, much variation), since these games are largely replicable and well-known. This is the one context where I think one might reasonably try to discern a language-specific effect in the language games used by speakers of that language. I have (had) not heard of an inversion transformation similar to French verlan used in English, though there is this on "back slang". But such transpositions are common enough (and not a general feature of natural language) that any gaps in the English-speaking world are probably accidental.