The reason it doesn't upgrade to a Keshik if you're not using the Mongols is because Civ3 (and PTW) skips over units in the upgrade chain if you can't build them. (For example, if you so desired, you could skip from Horsemen to Cavalry without discovering Chivalry.)
IIRC, the chain of horse upgrades is Horseman--Mounted Warrior (Iroquois UU)--Knight--Keshik (Mongol UU)--(all other UUs replacing Knights)--Cavalry--Cossack (Russian UU)--(all other UUs replacing Cavalry).