Misuse of the word "Timely"?
It strikes one as odd because timely is an adjective, and in OP's example the word to express how the reports are needed is expected to be an adverb. The mind (and the lexicon so far as I have been able to determine) recoil at the construction which might be formed by the usual practice of tacking -ly onto an adjective to make an adverb. Good grief, we'd end up with a word like timelily.
No matter, from north of the border I bring you tidings of the adjective timeous which OED declares to be
Originally Scottish. Now chiefly Scottish, Irish English (northern), South African, East African, and West African. A. adj. Categories »
- Done or occurring sufficiently early or in good time; prompt; (Law) done before a fixed time limit has expired. Cf. timely adj. 1c.
... even better timeous comes with the adverb timeously. Stick that to your coworkers.