Where does the apostrophe go for the word children in Mothers hold their children's or childrens' hands for a short time but their heart forever?

The plural possessive form is children's.

The plural form for the noun child does not add an -sto the end of the Word for the plural possessive.

Any plural noun form that does not end in -s, add the apostrophe -s to the end of the Word to form the possessive: men's room, oxen's yoke, mice's nest, etc.