The short answer is that at this point in Ref. 1 $u^m$ are new independent variables; not functions of $q$ and $p$. Equivalently, $u^m$ are Lagrange undetermined multipliers.
It is possibly helpful to mention that a field redefinition $u^m\to u^m +U^m(q,p)$ is allowed.
References:
M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994; subsection 1.1.4: eqs. (1.12) + (1.13).
P.A.M. Dirac, Lectures on QM, 1964; eqs. (1-7)+ (1-8).