What is the activation time of "Last One Standing"?

Is it an action? If so, it means that (on a successful activation roll) you use one action to activate it and the following action to eliminate all enemy minions in the encounter.

Stuff you choose to do and requires a check usually count as your action. I think you're right - you use your action to activate it then lose your next turn while you hop about in the background hacking people's limbs off.

I think most signature abilities are initially actions. Some of them have entries in the trees that lets you downgrade them to manoeuvres or even incidentals.

Last one standing is an action that puts you "out of action" for the following round as you mop up the mooks.

My understanding is that, once you activate the ability, the targeted minions are out of the encounter. Narrate appropriately through the two rounds that it takes to "finish," but don't have the minions take any more actions.