Simple null
Theorem: If regularity conditions (A)-(C) hold, then
\[\u(\bts) \tp \fI_n^{-1}(\bts) \u(\bts) \inD \chi_d^2.\]This follows directly from the distribution of the score.
Note that the consistency of the MLE is not required here since the MLE is not involved.
Nuisance parameters
Theorem: If regularity conditions (A)-(D) hold and $\bt_0=\bts_1$ (i.e., if $H_0$ is true), then
\[\u_1(\bth_0) \tp \fV_{11}^n(\bth_0) \u_1(\bth_0) \inD \chi_r^2,\]where $\bth_0$ is the restricted MLE.
The proof of this theorem is a homework assignment.