Let $n \in \mathbb N$. Given that $H_n(X_q) \cong H_n(X)$ for all $q > n$, the UCT tells us that \begin{align} H^n(X; G) &\cong \mathrm{Ext}(H_{n - 1}(X), G) \oplus \mathrm{Hom}(H_n(X), G) \\ &\cong \mathrm{Ext}(H_{n - 1}(X_q), G) \oplus \mathrm{Hom}(H_n(X_q), G) \cong H^n(X_q; G) \end{align} for all $q > n$.
Does this work? I have my doubts about this argument because the splitting in the UCT is not natural, but one of my cohorts said that this is the way to think about it.