Can I react with Tideturner on one battlefield, if another Showdown is happening on another battlefield?

Yes. You can play a hidden Tideturner from one battlefield as a Reaction to events during a showdown at another battlefield.

Tideturner played from hidden gains Reaction timing, meaning you can play it during Closed State (when something is on the Chain) or during Showdown Open State (when you have Focus). The showdown doesn't need to be at Tideturner's battlefield — you can play it from any battlefield you control.

Additionally, Tideturner's "When you play me" triggered ability is explicitly an exception to the normal hidden card targeting restriction — it can choose a unit "at another location," meaning it can swap with a unit at the battlefield where the showdown is happening (FAQ 7907, FAQ 1652).

One timing catch: If the showdown just began and nothing is on the Chain yet (Showdown Open), you need Focus to act. The player who initiated the showdown has Focus first. If you don't have Focus yet, you'll need to wait until a spell/ability goes on the Chain (creating Closed State + a Reaction window) or until Focus passes to you.