I recommend completing the speed run achievements in this order, following MxSvnty4's solution.
Egypt unlocks the Infinite Health cheat. Not only does this make you immune to enemy attacks, but you can walk through fire, traps, spikes and fall any distance without dying, so instead of climbing down you can just jump, saving precious seconds and making the following speed runs that little bit easier. Egypt also gives you the Infinite Mini SMG Ammo cheat.
Greece unlocks the All Weapons cheat. The only reason for doing this one second instead of third is so you can skip the whole section of Natla's Mines around getting your guns back. Greece also gives you the Infinite Shotgun Ammo cheat.
Lost Island with Infinite Health and infinite ammo is a breeze. Lost Island unlocks the Golden Shotgun (which fires 64 shells between reloads and kills with a single shot) and Silver Mini SMGs cheats.
Peru with Infinite Health and the Golden Shotgun is now substantially easier. You can even down the T-Rex with a single shot. Peru unlocks the Infinite 50 Caliber Pistol Ammo and Show Enemy Health cheats, but you don't really need them now.