Sustained turn rate wins nose-to-tail turning fights. Which simply means both guys turning the same direction. Which is most turn fights you see.
Which is why if someone says "Which turns better? Spit or Corsair? P-47 D-11 or Ta-152" to be honest one has to say "It depends" 
I wouldn't say sustained turn rate wins nose-to-tail fights. Consider an extreme example - suppose one pilot had a turn radius of zero, while the other had any non-zero turn radius. If they were engaged in a turning fight and pilot, the pilot with the bigger turn radius will NEVER be able to get his nose to point towards the guy with zero radius. It helps to imagine the guy with zero radius in the center of the circle.
In any less extreme case, if the circle of a plane with a smaller radius is completely inside the circle of a second plane with larger radius and both were committed to a nose-to-tail turn fight, the "outside" plane once again will NEVER be able to get his nose pointed towards the tighter turner.
Addendum: if the smaller radius plane is only partially inside the bigger radius turn circle, that means the circles intersect at some point and the bigger radius plane will have a chance for a snapshot.