At-sea engine rebuilding is not a complete machinery replacement

Can a navy change a warship engine while the ship is under way on deployment?

Extensive rebuilding can be done at sea, but that is not the same as removing and replacing an entire engine.

Navies have managed extensive machinery work at sea. The likely example involves engine reconditioning or rebuilding rather than a complete machinery replacement. HMS Invincible's southward voyage during the Falklands War may have included such work. Restoring machinery during deployment is a different industrial task from cutting open a ship for a full exchange.