Will nato go to war if Russia sends a few drones over a member state?

Of course not.

Nuclear powers would only use nuclear weapons if it's the last resort.

So when in the salami tactics world does it get used?

It's not exactly a new scenario:

Riots in West Berlin, buildings in flames. East German fire brigade crosses the border to help. Would you press the button? The East German police come with them. The button? Then some troops, more troops just for riot control, they say. And then the East German troops are replaced by Russian troops. Button? Then the Russian troops don't go. They are invited to stay to support civilian administration. The civilian administration closes roads and Tempelhof Airport.