Believe it or not, there is potential theoretical relevance. If the formation of adverbs like carefully from adjective stems like careful is a matter of lexical word formation (in other words, if it is derivational morphology, hence appropriately discussed in Chapter 18 of The Cambridge Grammar of the English Language), then we should expect to find arbitrary lexical exceptions, and the absence of lexemes such as *bigly, *littlely, *bluely, *oldly, *youngly, etc., is to be expected. Lexical word-formation is riddled with such gaps: take a look at the table I gave in this post, comparing the strikingly irregular pattern seen in horror / horrify / horrific / horrid with terror / terrify / terrific / *terrid and stupor / stupefy / *stupific / stupid and fervor / *fervify / *fervific / fervid and so on.
Odd set of examples. Bluely, oldly, youngly are well-established and not exceedingly rare, with good coverage in the OED.
Bigly was common enough a century ago but has since petered out.
Littlely as an adverb has a few OED cites but littly (adv) is the more common spelling.