This adjective isn’t my favourite. Perhaps it’s that weak, adverb-like —ly ending.
It’s unobjectionable in a timely reminder, but in a timely manner is like fingernails down a chalkboard, somehow.
I think that’s because of its fussy, needlessly formal tone and use of four words when one or two would suffice. On time (or even early) would do just fine.
Also unattractive is the American legal usage that turns the adjective into an adverb (as seen in the title of this US blog post): Does Your Company Timely Respond to All Reports of Potential Misconduct?
Are there are other ways to phrase it?
Promptly. That works.
Timeously. Very Scottish, so a bit outlandish in North America (or England).
Timelily. Perhaps the weakest of weak adverbs, with that unusual (but correct) —lily ending. Don’t.
I’d go with on time.
And in the timely reminder sense, you could also say well-timed, opportune, seasonable.