Here is an answer not referring to $\tilde G$ explicitly. I assume $G$ connected.

The given action $G\times M\to M$ defines a map $\pi_1(G)\times\pi_1(M)\to\pi_1(M)$, which can be restricted to $\rho\colon\pi_1(G)\to\pi_1(M)$. Geometrically, realize a class in $\pi_1(G,1)$ by a loop in $G$ and act by this loop on a point $*\in M$: you get a loop in $M$, which represents a class in $\pi_1(M,*)$. Well, it is more or less obvious (representing the points of $\tilde M$ as homotopy classes of paths in $M$) that the action lifts iff $\rho$ is trivial, $\operatorname{Im}\rho=\{1\}$.

Here is an answer not referring to $\tilde G$ explicitly. I assume $G$ connected.

The given action $G\times M\to M$ defines a map $\pi_1(G)\times\pi_1(M)\to\pi_1(M)$, which can be restricted to a homomorphism $\rho\colon\pi_1(G)\to\pi_1(M)$. Geometrically, realize a class in $\pi_1(G,1)$ by a loop in $G$ and act by this loop on a point $*\in M$: you get a loop in $M$, which represents a class in $\pi_1(M,*)$. Well, it is more or less obvious (representing the points of $\tilde M$ as homotopy classes of paths in $M$) that the action lifts iff $\rho$ is trivial, $\operatorname{Im}\rho=\{1\}$.