If I have Brynhir Thundersong ready in my base, then I play Temporal Breach. Will my Brynhir Thundersong get play again ready?

No — Brynhir is banished and then replayed, so she comes back to Base exhausted, not ready. A unit only enters ready if something like Accelerate or an "I enter ready" effect applies; Temporal Breach just plays her normally, so the default applies (rule 359.2.c).

Do note her "When you play me" trigger fires again, so opponents can't play cards for the rest of that turn (and they can react to that trigger before it resolves).

One caveat on the setup: played from hand, Temporal Breach can target your Brynhir at Base fine. If you were trying to do this with a Hidden Temporal Breach, it could only banish a unit at the battlefield it was hidden at — never a unit at your Base (CR 811.1.d.2).