If I activate Overzealous Fan ability, can I react with a Hidden Tideturner in the other battlefield to change with Fan and Fan continues pushing back the unit in first battlefield?

Yes, you can react with a hidden Tideturner to swap it with Overzealous Fan, and Overzealous Fan's ability will still resolve successfully.

Sequence of Events

1. Overzealous Fan's Trigger: When Overzealous Fan gains the "Defender" designation, its "When I defend" ability triggers and is placed on the chain. You must choose the target (the attacking unit) at this time.
2. Reaction: You play Tideturner from hidden as a reaction to the Fan's trigger.
3. Tideturner Resolves: Tideturner enters the battlefield and its "When you play me" ability triggers. You swap Tideturner with Overzealous Fan. Overzealous Fan is now at the location where Tideturner was hidden.
4. Fan's Ability Resolves: When the chain reaches the Fan's ability, you make the decision of whether to kill the Fan to move the attacking unit. Because the Fan is still on the battlefield, it is a valid target for its own "kill me" cost.

Why this works

• Valid Target: According to Rule 356.3.e.6, instructions that cannot be followed are ignored. The Fan's ability does not require the Fan to remain at the original battlefield to be killed; it only requires the Fan to be on the battlefield. Since the Fan is still on the board after the swap, the cost "kill me" can be paid.
• No Whiff: Unlike the scenario in FAQ 916, where the Fan is returned to the hand (a non-board zone) and thus cannot be killed, swapping the Fan to a different battlefield keeps it on the board. Therefore, the ability does not whiff.
• Targeting: The target of the Fan's ability (the attacking unit) was locked when the ability was placed on the chain. Moving the Fan does not invalidate the target of the ability.

Note: I could not fully verify this ruling against the retrieved rules.