Will changing the diameter or length of the axle change this relationship?
You should think of torque as the rotational equivalent of a force. When you apply a force to one end of an object (i.e., you push it), the other end also moves. If this didn't happen, you would be compressing the object, squeezing it, as it were. The same happens with torque. If the object is solid and doesn't deform, the torque will be the same everywhere. If the object is made of play-dough, for example, the torque is used up is deforming the object instead, and almost none of it reaches the other end.
Changing the dimensions of the axle will, however, change its moment of inertia. Since the angular acceleration that a given torque imparts on an object depends on the object's moment of inertia, changing the dimensions of the axis will change the angular acceleration that it feels. So, for example, the axle could start speeding up a lot more, or it could feel harder to spin, depending on how you change it.