Let $G$ be a locally-compact topological group. Let $C_0 (G)$ be the Banach algebra of the continuous functions that vanish at infinity and let $M(G)$ be its dual, which can be identified with the space of all regular Borel measures of finite total variation. Endow $M(G)$ with the weak* topology.
Consider the convolution $M(G) \times M(G) \ni (\mu, \nu) \mapsto \mu * \nu \in M(G)$. I can prove that it is separately continuous, and if $G$ is compact I can prove that it is jointly continuous.
Is it possible to change the above framework (for instance, the space of functions considered or the topology on it) in order to obtain joint continuity for more general groups?
An answer to the question in the title, which is not quite the same as the question in the body of the post:
Presumably it's possible to "change the framework" as suggested to obtain joint continuity for more general groups. In case you didn't realize, in the setting of your question, compact groups are the only groups where you get joint continuity:
Say $G$ is not compact. Let $(x_\alpha)$ be a net in $G$ that tends to infinity.
(That is, for every compact $K$ there exists $\beta$ such that $x_\alpha\in G\setminus K$ for all $\alpha>\beta$; you can construct such a net by using the compact sets themselves as the index set, letting $x_K$ be any element of $G\setminus K$.)
Let $\mu_\alpha=\delta_{x_\alpha}$, $\nu_\alpha=\delta_{-x_\alpha}$. Then $\mu_\alpha\to0$ and $\nu_\alpha\to0$ but $\mu_\alpha*\nu_\alpha=\delta_0$.