The mechs are not going to jump.
@Arx: I'm aware that we're not talking about small arms, and by making that observation you're not doing anything to convince me that you could shift 60 tons with projectiles massing 20 kilograms. Artillery shells are either going through, detonating against, or they are bouncing off the armor. Artillery shells do not PUSH tanks.
I'm still not under the impression they're bipedal, and if they're to function at all, they have to have decent amount of surface area for their feet anyway, or some sort of handwavy science magicks.
They are bipedal. I am perfectly happy handwaving ground pressure if that's the only offense it takes to get mechs and medieval people fighting side by side. So let's just gently skirt that issue rather than try to be original or solve it.
Hey, I got here a bit late, but I have some ideas. First, don't havewave ground pressure! Not entirely, anyway- medieval architects understood this principle well and if they didn't they learned fast (see the Leaning Tower of Pisa.) If we just assume a mech can stand on dry soil fine (not too much of a stretch, you could even say it leaves deep footprints) that leaves room for it have difficulty navigating deep mud, marshes, and bogs, which is a problem also suffered by heavy cavalry. If non-mech-using armies can get a much to retreat or chase them into poor terrain, it could be immobilized or slowed down, a tactic also employed against heavy cavalry.
You haven't settled on any details of how the mechs are "dug up" have you? I think it makes the most sense that they would be stored inside some kind of bunker facility, which is itself in a state of decay. Perhaps it was only discovered because a section of roof collapsed, which caused a sinkhole on the surface. This readily means that mechs could have suffered varying degrees of damage before even being found. You can explain the absence of any technology that one would expect to find in a bunker by its being collapsed upon and destroyed, and you could have it be unearthed whenever you like. Maybe there's a door leading to a deeper, more intact level of the bunker with something game-changing like a fully intact mech(s), a supply of ammunition that had previously been extremely scarce, or a WMD like a nuclear bomb or bioweapon. You could even have them find the WMD in the final act and use it, only to end up wiping out both themselves and the enemy almost entirely from the fallout/spread of disease/whatever.
The engine should definitely be some kind of nuclear reactor. Fuel rods are loaded in, coolant heats and moves through a turbine or whatever, then coolant cools off in a heat sink. I'm not sure how long refined nuclear fuel lasts, but I'm guessing there's way to store it (IE with the individual pellets isolated from eachother most likely) so that it lasts as long as unrefined nuclear material deposits in the ground last, which is to say, millions of years. Fuel loaded into a mech would probably last months or years but someone has to take the fuel out of storage and load it into the mech first. As soon as it's discovered that handling fuel rods causes you to break out in tumors and die, this task will fall to some poor group of serfs/slaves/prisoners/"volunteers". If for whatever reason to coolant leaks out, it could be replaced with ordinary water, but would be far less efficient. Maybe pilots/crews who push their mech too hard could also run the risk overheating and damaging components or forcing an automated shutdown. You could have some mech crew member heroically sacrifice himself by performing a mid-battle fuel rod change
I would probably skip on the sentient AI. It's a little far into 'soft' sci-fi and you'd also have to handwave the language barrier somehow.
Regarding how mechs could be defeated, I think it would be neat if it was possible to somehow scale a mech and kill its pilots. This would be viable if mechs lack point-defense weapons, and would make the mech dependent on infantry support, etc. Attackers would need to be able to enter the hatch/cockpit though- if mechs have glass cockpits, maybe some of them were broken by bunker collapses. Alternatively, whatever the means of entry, maybe the people using the mechs had to force their way in and damaged the locking mechanism in the process, especially if it's something complicated like a computer-controlled maglock. If there's ever mechs on both sides then one would assume the mechs have weapons that are effective mech-to-mech even if they're immune to medieval cannons, but ammo is probably precious scarce and if there are DEWs they depend on the reactor being well fueled and in working order. Tripping up the mechs with ropes (or more likely, iron chains) Empire Strikes Back style might or might not be plausible depending on the scale and mass of the mechs. If there's bigger and smaller ones, maybe the small ones are capable of being tripped, but big ones can lift twenty horses up by their own ropes and throw them across the battlefield like toys.
Any "new" mechs would probably have to be wheeled, powered carriages. If we assume that controlling the walking mechs requires a strong amount of computer automation (which makes sense) then it would be outright impossible to reproduce. If refined fuel is already available though, then dangerous and inefficient nuclear reactors are probably fair game, and they can be put to use on wheeled war engines. Powered ships would be even easier to engineer. Given the relative scarcity of iron, the labor-intensive process of making good-quality steel for armor and weapons, then it is very unlikely this medieval society could make mobile armored vehicles capable of deflecting large cannonballs, but the crew could be protected from small arms. Maybe, if mechs are every tripped/bound with chains, it requires the use of a few of these motorized vehicles. There's also room for ad-hoc use of salvaged mech parts, for example the weapons could be placed on carriages or used as static defenses atop castle walls. Unless only the static variety are being used, it would probably put an end to siege warfare and the age of castles in a very short time.