In RL any yellow(ish) dye would wear out first as it has the least 'mordant' compared to other colours. Witness the almost blue Dragoon and Chasseur Napoleonic uniforms in museums. So while looking good, is is doubtful that anything red would fade to orange as there isn't any or hardly any yellow dye in the mix. ( except for the bonding agent)
So it depends a bit on which dye was used to get the red colour in the first place. The most popular in the 19th c. was iron Oxide which fades to brownish pink after prolonged use. The other one is Red Ochre which will fairly rapidly turn to an almost lilac- pink. Wool tinted with madder roots turns greyish pink. Hopes this helps somewhat.