'A random direction is picked from the NESW plane and you're pushed in it. Since nil space is anisotropic, how much do you go up or down?' This question doesn't actually make sense because you cant say whether two points are up or down relative to each unless they are on the same vertical axis. When you say the bird immediately goes up, thats just how it looks like due to the projection to euclidean space. The bird isn't really 'up' or 'down' until they make a loop, at which point it is displaced by the area of the loop.
When you choose an arbritray coordinate system, then it stop being isotropic I think, because then you can apply your argument, but the coordinate system is not an intrinsic part of the Nil geometry
You should watch Zeno's video 'Nil geometry explained' if you haven't already.