Depends on the lead alloy.
A typical alloy using wheel weights or other source of antimony for hardness that also has a very little bit of arsenic in it (like 0.01% total arsenic in the lead alloy - and almost all antimony and many sources of lead will have some arsenic in it) that gets dropped from the mold into water (essentially heat-treating the bullet) will take a couple days to reach full hardness, then after a few years will start to soften and after a decade or more will reach a stable hardness that won't change any more.
If dropped to rag pile and allowed to slowly air-cool the bullets won't harden, but then won't soften either.
Pure lead-tin alloys don't exhibit this behavior, you need the antimony & arsenic for heat-treating to be possible.
It's been discovered that it's the antimony & arsenic together that allows lead alloys to be heat-treated for extra hardness, but the tempering temperature is very low and the bullets will very slowly anneal over time at normal room temperatures. Several good posts about this over at
Cast Bullets forums.