Lead ain't that dangerous. Just take it out and dispose of it like you do with normal batteries. Clean your hands afterwards and you're dandy. As for the clock, the battery contacts, and whatever they were attached to, are likely eaten away, but I can't say that for certain from this photo. If you're lucky and they're mostly intact, some IPA scrubbing and a dip in vinegar, and a bit more scrubbing, should take most of the crust away. That rust though, probably some vinegar, maybe a deoxidating agent (like navy jelly?) could clean it off. Even cleaning all of it doesn't guarantee that it'll work any way.

An alternative would be to replace the movement.

Could be easier than fussing with the contacts.

After you have cleaned it all (as the other comments explained already) and put a new battery in there, it is possible that the electric contacts are still not clean enough (especially the left one with that deep rust) and it doesn't work with the new battery.

Then you have to go at the contacts with a small file or a small piece of sanding paper and scratch them very good, so that they get really very shiny right there in the middle, where it counts. Then it will work again.

Just don't lick the residue and you'll be just fine

As long as the decorative side isn’t affected, it should be fine (and that’s only an esthetic thing anyway).

Just clean it up and try to get as much residue out of the back as you can. The easiest thing to do is just throw out the whole clock mechanism and replace it (very, very easy), but if you don’t want to do that, and the mechanism itself isn’t affected, you could probably just buy a new battery enclosure with the same battery capacity as the original, and wire it to the same connections the original batteries had.

Just clean it up. Wear gloves and a mask to prevent anything getting in your body. Just be careful about what cleaning products you use to prevent releasing dangerous gasses and do it in a ventilated room or outside. But I would recommend that for any battery or strong acid or base or other reactive chemicals, not just lead containing ones.

Well, it's not going to explode because it already has...

But if it's really so old that it's lead you should be careful. Is it an antique clock where it has value sentimental or other? If not just toss it.

The mechanism is pretty standard, so if you really want to salvage it, you can replace that (and keep the same hands) but definitely mask up and take it outside to clean the corrosion off. If it's an antique then this has already happened at least once.