"the cuckoo bird lay her egg in the nest. " That's exactly how the cuckoo chick 'invaded' the nest. Because its mother laid its egg in another bird's nest.
"when the snake ate the chick the eggs are still in the nest." You missed the 1st egg kicked out by the cuckoo chick at the start of the video. The snake ate the cuckoo chick before it could kick out the remaining 2 eggs.