I use Kiwi shoe polish (paste) and Kiwi leather dye (in an applicator bottle) both in black to touch up my Alessi gear. I believe it was Lou that suggested the Kiwi polish (color to deal with scuffs and Neutral to finish) and I had the dye handy so I tried it on the really worn places.
Kj, Shortround gave you great advice. Get a small bottle of dye to match the edge finish on your belt, and just carfully daub some dye along the edge. When that's dry, then use a good wax like Kiwi shoe polish to finish the edge. It should look glassy, and smooth.