This is an interesting variation of First passage percolation (see [this question][1]). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are.

I think that the speed should be strictly larger than $\sqrt{2}$, for the following reason: if we had a path using NE diagonals almost all the time with high probability, then the graph consisting of only the long (say >100) streaks of NE diagonals should percolate. But standard percolation argument should show that this isn't the case. I hope this makes sense, I don't have time to check the details right now.

ADDENDUM: regarding fluctuations we have the paper of [Benjamini, Kalai and Schramm][2]. There's a newer paper by [Benaïm and Rossignol][3], which I haven't read. I guess that most of the results apply to this setting as well.


  [1]: https://mathoverflow.net/questions/9558/the-shortest-path-in-first-passage-percolation
  [2]: http://front.math.ucdavis.edu/0203.5262
  [3]: http://arxiv.org/abs/math/0609730