Well, “timely” as an adjective doesn’t really mean “quickly”. It comes closest to meaning “fortuitous” in a chronological context. “You arrival was timely” doesn’t mean, “You got here quickly.” It means “Your arrival at that particular moment was fortuitous.” It’s actually a fairly complex adjective when you analyze it.
Of course, but an adverb formed with “timely” could be useful as well in some cases. “He arrived in a timely manner” = “He arrived <…>”. I guess we could say “He arrived opportunely”, but that sounds wrong, or to use your suggestion, “He arrived fortuitously”, which I figure is right but is rather long. Maybe there just isn’t any adverb that can be formed from this adjective.
“. . . he arrived punctually.”
Tris
Even if you add -ly to an adjective it doesn’t always result in an adverb. “Shortly” means “soon”. It does not mean “in a short manner”.