If $f$ is a newform in $S_k(\Gamma_0(N))$ then $f$ is an eigenfunction for both $U_q$ and $W_N$. But in general $U_q$ and $W_N$ do not commute. You can find an example in Shimura, "Introduction to the arithmetic theory of automorphic functions", Remark 3.59. There he constructs eigenfunctions $f$ for $U_q$ such that $W_N f$ is not an eigenfunction for $U_q$, hence $W_N U_q f \neq U_q W_N f$.

EDIT. In the answer below, $U_q$ refers to the usual Hecke operator given on Fourier expansions by $\sum_{n \geq 1} a_n x^n \mapsto \sum_{n \geq 1} a_{qn} x^n$. The operator $U_q$ in the OP is given by $\sum_{n \geq 1} a_n x^n \mapsto q \sum_{n \geq 1} a_{qn} x^{qn}$. As explained in the comments this does not preserve the space of forms of level $N$.

If $f$ is a newform in $S_k(\Gamma_0(N))$ then $f$ is an eigenfunction for both $U_q$ and $W_N$. But in general $U_q$ and $W_N$ do not commute. You can find an example in Shimura, "Introduction to the arithmetic theory of automorphic functions", Remark 3.59. There he constructs eigenfunctions $f$ for $U_q$ such that $W_N f$ is not an eigenfunction for $U_q$, hence $W_N U_q f \neq U_q W_N f$.