Most of the forms you cite are literary.

The word order used in first one (le falloir) used to be the standard until the 18th century when the usage moved to the current one (falloir le).

Il le fallut mettre en nourrice → Il fallut le mettre en nourrice (Sending him to wet nurses was necessary)

See this NGram:

The remaining sentences do not use this archaic inversion but a different formal pattern: subject + pronoun + falloir + infinitive.

The pronoun (me, te, lui, nous, vous, leur) is an indirect objet representing who is required to do something. E.g.:

Il me faut partir.

The usual way to express the same is:

Il faut que je parte.

The second sentence moves also the verb at the end of the sentence while current usage would not do it :

qu’il lui faudrait tout à l’heure abandonner → qu’il lui faudrait abandonner tout à l’heure

The remaining sentences follow the same formal pattern:

Il lui fallait attendre → il fallait qu’elle attende (feminine here because elle se levait en sursaut)

Il fallait attendre → "To wait was required"

Il lui fallait attendre → "It was a requirement for him to wait"

This is different from the first sentence where le refers to the object (him) while here lui refers to the subject (for/to him). Note that this usage is not literary when falloir is followed by a noun, for example :

Il lui fallait une heure = He/she needed one hour.

Compare to :

Il fallait une heure = One hour was needed.

Here is an interesting reference from Jean-Michel Kalmbach about falloir usage .