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This question is about learning to speak a language as a mother tongue (i.e., from birth (L1), not as a second language (L2)).
Some languages have more grammar rules than others (number of cases, number of tenses, etc). Is there any evidence for or against the hypothesis that a more complicated grammar would make a language take longer to learn as a mother tongue (L1)? "Longer to learn" could manifest as delayed speaking or speaking more simply for a longer time.
The article
Bleses, D., Vach, W., Slott, M., Wehberg, S., Thomsen, P., Madsen, T. O., & Basbøll, H. (2008). Early vocabulary development in Danish and other languages: A CDI-based comparison. Journal of child language, 35(3), 619-650.
compares several studies that measured how fast children acquire languages by various measures. It argues that languages with plenty of vowel sounds are harder for children to learn. Some comparable languages in the study are Danish and Swedish, and UK and USA English. Danish was harder than Swedish, and UK English harder than USA English.
Of these, the Englishes have similar grammar, while Danish has, at least on surface, simpler grammar than Swedish.
This would suggest that grammar is not a deciding factor on how difficult a language is to learn as a child.
A more throughout answer would use an actual measure of grammatical complexity and check the included languages for that.
Anyone interested in the subject is encouraged to read the paper and see if there are more recent studies in the same vein; Searching for the article on Google scholar and seeing what refers to it would be a good place to begin.
In lieu of official studies, here is an observation... Let's take English language as a language with simple grammar, and Russian language as one with complex grammar. If we distinguish "spending more time to learn grammar" from "communicating complex ideas at a different age" then we can compare learning English and Russian by native children as following:
However, if we look at what stories are read by kids of the same age from both backgrounds, we will find that they read books of the same complexity.
Conclusion: Children using a more complex language spend more time learning its grammar. However, it takes much longer for a child's brain to develop in order to understand and express more complex ideas than it takes to learn grammar.
TL;DR It takes slightly longer.
You might want to check out http://psychology.wikia.com/wiki/Toddler. It has useful information, such as the facts that 10-20 words can be taught. We all know that a child's brain development hasn't grown, yet it hinders their education, right?
Languages like Japanese for example have more grammatical aspects to it. It has polite and standard forms, and tenses based on animate and inanimate objects. Remember we're talking about someone's mother tongue! Children have absolutely no concepts of grammar when first learning it, they can't say "私 (watashi)" means I in English.
So, it's very obvious. Yes, you didn't want knowledge of things to do with second languages but here's one interesting thing. Now, the FSI (Foreign Service Institute) has Japanese as Category V (it means it takes about 2200 hours I think when I last checked), opposed to languages without rather complex grammar ideas.
There is likely going to be a psychological study, of course. However, it's common sense and checking out school curriculums are quite good.
