Plural "concatenated compound nouns" such as mothers-in-law, courts-martial, fillers-in, passers-by contain a "head noun" - they're either clearly-identified types (of mother, court,...), or they're nouns followed by a preposition (in, by,...) that "contextualizes" the relevant noun. See here for more details.
It's something of an archaic hangover that with many such terms we still usually pluralize the head noun, rather than the whole expression. I've no doubt some people go out of their way to remember and make use of the pluralize the head noun principle primarily to show how "educated" they are. And there will be others who know perfectly well which is the "correct" form - but deliberately flout it to show contempt for those few usages that are out of step with the standard method (add s to the end of the complete noun phrase, regardless of any internal "structure").
Many people recognize the principle that words ending in -ful use the modern standard plural syntax (teaspoonfuls, handfuls) - but Google Books contains thousands of written instances of two teaspoonsful, handsful,... - which just goes to show that when the rules get too complicated, compliance falls off.
See this NGram showing how "pedantically correct" poets laureate is gradually being replaced by "naturally conformal" poet laureates. It's a long slow process, but the change is mostly in one direction.
In the case of forget-me-nots, merry-go-rounds,... there's no obvious head noun, so no-one is tempted to put the pluralising s anywhere other than at the end.
Finally, here's an interesting discussion re pluralization of Portuguese man o’ war (a type of jellyfish), where at least one "competent writer" chose to use the singular as the plural - the way we do with sheep and deer.