However, libsecp256k1 takes its nonce as input to its API, and from that point on signing and verification are deterministic functions. Any nonce skew would need to occur in the Bitcoin code which calls into libsecp256k1; however, since November nonce generation has been deterministic as well (using RFC6979). This code has been audited and replicated by myself and others; it is also unit tested.
This is not technically true anymore. Since recently, there is a full RFC6979 implementation inside libsecp256k1, with test vectors that were generated by another implementation (feel free to review it; it's too recent to go in Bitcoin Core v0.10.0 still, though). The reason for this change was making sure that the easiest way for using the library is always safe - the old API allowed you to shoot yourself in the foot if you passed in a bad nonce.