Let $G$ be a topological group. $G$ comes equipped with a left (resp. right) uniformity $\mathscr{L}$ (resp. $\mathscr{R}$) which can be characterized as the coarsest uniformity which is compatible with the topology and which makes $x \mapsto gx$ (resp. $x \mapsto xg$) a uniformly continuous map $G \to G$ for all $g \in G$.
Edit: My question is now just:
Is there necessarily a uniformity on $G$ compatible with the topology which makes all left and right multiplication maps uniformly continuous? Bonus points if multiplication $G \times G \to G$ (using the product uniformity on $G \times G$) is uniformly continuous or inversion is continuous.
As Harry Altman points out, there must be (as for any uniformizable space) a finest uniformity $\mathscr{U}$ on $G$ compatible with the topology. Since the uniformities on $G$ form a (complete) lattice there is also a coarsest uniformity $\mathscr{V}$ refining both $\mathscr{L}$ and $\mathscr{R}$. Any uniformity which answers my question must sit between $\mathscr{V}$ and $\mathscr{U}$. Such a uniformity is automatically compatible with the topology since it will sit between, say, $\mathscr{L}$ and $\mathscr{U}$ which are compatible with the topology.