Let $G$ be a locally compact group with Haar measure $\mu$. Is right-translation continuous on $L^1(G,\mu)$?

Question

Let $G$ be a locally compact group with right-invariant Haar measure $\mu$. I know that the space of compactly supported continuous functions $C_c(G)$ is dense in $L^1(G,\mu)$. If $G$ is 1st-countable, I can show that whenever $y_n\to y$ in $G$, we have that for any $\varphi\in C_c(G)$, if we let $\varphi_{y_n}(x)=\varphi(x y_n)$, then $$\varphi_{y_n}\xrightarrow[L^1(G,\mu)]{n\to\infty}\varphi_y$$ by dominated convergence, so that the map $$G\to L^1(G,\mu),\quad \phi\to\phi_y$$ is continuous. But is this still true if $G$ is not 1st countable?