The stakes of a Conflict aren't really determined until someone concedes or is taken out. This means that the nature of a conflict can change, story-wise, without there being any need to change anything game-wise. You just keep running Exchanges, letting people narrate and roll for whatever tactics they want to employ, until the Conflict ends, and that is the point at which we learn exactly what kind of Conflict it was, by the winner declaring what their victory means in story terms.
Ending a Conflict and starting a new one isn't the right way to address changes in the action or strategies, because ending a Conflict resets Stress tracks. That doesn't make sense if the Conflict really has not ended and is really still going on, but in a different way.
Rando provides an excellent answer about changing from a Conflict to something else that's not a Conflict at all, and also made the excellent point that "you're continuing the conflict" if something changes but they're still trying to take each other out.
Last thought: It's not really right to say that "Conflicts are either mental or physical in nature", because it absolutely is RAW for both social/mental and physical attacks to all be part of the same Conflict. It's not ideal to switch mid-fight, because if you give a foe the opportunity to soak Stress on not one but two Stress tracks, that means that it's going to take longer/more to take them out, but there's nothing special in the rules about switching. It isn't ending one Conflict and starting another one, it's employing different tactics in the same Conflict. (Fate Core page 160 makes no statement at all requiring a Conflict to be only all-physical or only all-mental.)
If you take Stress from a physical source, you check off a physical stress box. If it's a mental hit, check off a mental stress box.
Nothing at all about one stress track or the other being ruled out due to some notion of an overall Conflict type.
Once again: The most important thing in a Conflict is, what's at stake. Even this can change as the Conflict unfolds: People might have stakes in mind, but the stakes aren't manifest until the Conflict ends with a Concession or a Take-Out and the winner states their stakes.