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Besides the position of ramification points $z_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a conformal automorphism of the sphere $L^1$.

No general formula exists. To find the uniformization in the case of genus $0$, you have to find a rational function with simple critical points and prescribed critical values. The degree of your function is $M+1$ by Riemann-Hurwitz formula. You write it with undetermined coefficients, add normalization, for example $f(0)=a,f(1)=b,f(\infty)=c$, where $a,b,c$ are some points distinct from your $z_j$, and write the condition that critical values are $z_j$. You obtain $2M+3$ algebraic equations with $2M+3$ unknowns, this system has finitely many solutions. Different solutions correspond to different possible monodromies. The system is difficult to solve when the degree is high.