When we had infants I'd go back to sleep pretty well instantly (co-sleeping, with baby on a side-cot at bed level; baby breastfed, not by me) but in general one would wake to the little snuffles that precede the crying (I guess that helps to calm them before they get in a tizzy [in a state, brought on by their own actions, crying in this case]).

I sleep through thunderstorms without stirring.

I'm a few years out from dealing with crying babies and still if there's crying in the background, that sounds like one of my kids, I'll pop-up like a meerkat. The sound even gets though (non noise-cancelling) headphones when I'm gaming, though I won't know what I've heard until I take them off.