Constant Regressor in GLS
The equation-wise OLS estimators are consistent under the further normalization that $E(u_{i1})=0$ and $E(u_{i2})=0$, which are naturally assumed to be satisfied. For GLS, I don't know what you mean by GLS, but if you mean the SUR estimator, then I see no reasons why inconsistent. The fact that $E(x_i' u_{i1}) \ne 0$ is irrelevant because $x_i$ is not a regressor in the first equation.