If I play a card and opponent reacts with Temporal Breach to replay Brynhir Thundersong, does my card still get played?

Yes — your card still resolves. Brynhir's "When you play me, opponents can't play cards this turn" only blocks new plays; it never counters anything already sitting on the chain, and your card was played before she was replayed (FAQ 13003).

Sequencing: Temporal Breach resolves first, Brynhir comes back as a new object and her play trigger goes on the chain above your card, so the lockout goes live before your card resolves — but your card still finishes resolving normally. You just can't play anything else that turn afterward.

One correction on the setup: Temporal Breach has no [Action]/[Reaction], so your opponent can only spring it this way from face-down (Hidden grants Reaction), which means Brynhir had to already be at the battlefield where the Breach was hidden to be a legal target (CR 811.1.d.2). Also worth knowing that if the card you played was a unit, there'd be no window at all — units finalize and leave the chain immediately, so there's nothing to react to unless a trigger from it is on the chain.