For large $q$ such a result does not hold. This follows from the results in the paper "Limitations to the Equi-Distribution of Primes I" by Friedlander and Granville. Specifically, their Proposition 1 implies that for any constant $B>1$ and arbitrarily large $Q$ (assuming GRH, all sufficiently large $Q$ work), if $q$ is a number in the range $\left(\frac{Q}{(\log Q)^{1/8}},Q\right)$ with no prime factors below $\log q$, then for some $x\in(Q(\log Q)^B,Q(\log Q)^B)$ and $a$ relatively prime to $q$ we have $\psi(x;q,a)-\frac{x}{\varphi(q)}\neq o(x)$.

It follows that for any $A\geq 1$, the uniform estimate $\psi(x;q,a)=\frac{x}{\varphi(q)}+o(x)$ cannot hold for all $q$ satisfying your condition. In case you are curious about $A<1$, it is impossible for a number $q$ to be a product of distinct primes smaller than $\log q$. Indeed, if $q$ is a product of $N$ primes below $\log q$, we have $N\leq\log q\leq\log 2^N<N$.

For large $q$ such a result does not hold. This follows from the results in the paper "Limitations to the Equi-Distribution of Primes I" by Friedlander and Granville. Specifically, their Proposition 1 implies that for any constant $B>1$ and arbitrarily large $Q$ (assuming GRH, all sufficiently large $Q$ work), if $q$ is a number in the range $\left(\frac{Q}{(\log Q)^{1/8}},Q\right)$ with no prime factors below $\log q$, then for some $x\in(Q(\log Q)^B,3Q(\log Q)^B)$ and $a$ relatively prime to $q$ we have $\psi(x;q,a)-\frac{x}{\varphi(q)}\neq o(x)$.

It follows that for any $A\geq 1$, the uniform estimate $\psi(x;q,a)=\frac{x}{\varphi(q)}+o(x)$ cannot hold for all $q$ satisfying your condition. In case you are curious about $A<1$, it is impossible for a number $q$ to be a product of distinct primes smaller than $\log q$. Indeed, if $q$ is a product of $N$ primes below $\log q$, we have $N\leq\log q\leq\log 2^N<N$.