Learn Hardcore Japanese: zyoukyou wo kakuninsite kara, taisaku wo kangaemasu. - After checking the situation, I will think of measures.

It means “after doing A, then (do) B,” with a strong sense that A must be completed before starting B. So 状況を確認してから = “after checking the situation (and only then) ….”

Can I use 〜た後で instead of 〜てから? Any nuance difference?

Yes: 状況を確認した後で、対策を考えます。 Both mean “after.”
Nuance:

〜てから emphasizes ordered, immediate succession and completion of the first action.
〜た後で is more neutral about timing and can feel “after that, at some later point.”

Is the から here the same から that means “because”?

No. This is Vて + から = “after doing.” The “because” から attaches to a plain-form clause, e.g., 状況を確認するから、安心してください。 (“Because I’ll check the situation, please don’t worry.”)

Why is を used after 状況 and 対策?

Because both are direct objects of transitive verbs:

状況を確認する (to check/confirm the situation)
対策を考える (to think up/consider measures).
Verbal nouns like 確認 behave like nouns that take を with する.

How do you read each word?