Theorem 2.6.6 in “USCO and Quasicontinuous mappings”, vol. 81, Berlin, Boston: De Gruyter, 2021, by L.Hola, D.Holy and W.Moors says (more-or-less): 

If f:HxX->G is a separately continuous mapping, H is a Cech-complete topological group, X is a q-space, G is a topological group, and for each x in X, h -> f(h,x) is a group homomorphism, then f is jointly continuous on HxX.

But perhaps this is too general for your purposes.