prep problem

on the way
is an idiomatic expression;
means (1) in the process of coming, going, or traveling (正在來,往);
or, (2) on the route of a journey (在途中).

"on the way" (or "on one's way") functions as adverb to the tell the place or location of the subject. You can extend the expression by putting another adverbial after it to add more information about the place or location.

"on the/one's way" + adverbial
example:
on my way home (adverb)
on her way out the door (prepositional phrase as adverbial)
on the way to town (prepositional phrase as adverbial)

Your sentences:
Both (1) and (2) are wrong.
It should be rewritten as:
We were caught in incessant rain on the way to school.

(3) on our way home ==> 在我們的回家途中

在我們的返學途中 should be "on our way to school"