'How do you notate melody notes?'
Well, I certainly don't use Roman numerals. They are kept for harmony and chords only.
I might use Arabic numbers, with approriate sharps or flats before (or after) them.
I might use solfege, whiich works well in movable doh, where doh is the root of the key.
I might use fixed doh, where doh is C always. This can become messy with every note having its own name, and often incurs extra 'dieze' or 'bemole' being attached to notes, which does make it tricky when singing those notes.
All of which won't have any timings shown.
So, the simplest, for me, since that's what you ask, is to eschew those and go for the old-fashioned dots on staves. Which actually can be read by any muso, anywhere, with no confusion. I.e. it works!