On the prime number theorem in arithmetic progression

For $x\leq\phi(q)$ the estimate $\pi(x,a,q)\ll\frac{1}{\phi(q)}\frac{x}{\log x}$ would imply $\pi(x,a,q)\ll\frac{1}{\log x}$, i.e. $\pi(x,a,q)=0$ for large $x$ which is clearly false. So a bound you envision can only hold for $x$ slightly above $\phi(q)$. On the other hand, for any $\epsilon>0$, the Brun-Titchmarsh inequality implies $$\pi(x,a,q)\ll_\epsilon\frac{1}{\phi(q)}\frac{x}{\log x},\qquad x>q^{1+\epsilon}.$$

Siegel - Walfisz theorem states $$\psi(x,q;a)=x/\phi(q)+O(x/\log^Ax)$$when q is small;q is big ,the results is trival . (When is the Siegel-Walfisz theorem non-trivial?)

if $q \ll log^Ax$,we have $$\sum_{x\equiv 1 \bmod q } \Lambda(n)\ge (1-\epsilon) x/\varphi(q).$$

(Prime numbers in arithmetic progressions : uniformity with respect to the modulus )

I think q is big, there is no the $\ll$ results as you asked. Since when q is big ,there may be existing Siegel-zeros, then there exist constants 0 < β−, β+ < 1 such that $x^{\beta_{-}}/\beta_{-}\ll x-\phi(q)\pi(x,q;a) \log x \ll x^{\beta}_{+}/ \beta _{+}$. (see section 3 , and http://www.dms.umontreal.ca/~andrew/PDF/ItalySurvey.pdf) or see Theorem (Siegel-Walfisz with a twist). in (Chebyshev function in arithmetic progressions)