I understand that native speakers of English would not say, "I picked up it."
I think "I picked it up and took it home" is correct.
Is the following sentence grammatically correct?
"I picked up and took it home."
pick up sg (sg = not pronoun)
pick sg up
pick it up
pick up it
When the object of the pv is a pronoun, always split the pv. Not a pronoun, then the separation is optional.
I picked up !! and took it home. you cannot drop the object of the phrasal verb
I picked it up and took it home.
I picked it up and took !! home.