In *B. Mitchell. Theory of categories* (1965) three properties of abelian categories were introduced.

An abelian category is termed $C_1$ if it is cocomplete with exact coproducts, $C_2$ if it is bicomplete and natural transformation $\eta_{-, X}: \coprod_X (-_x) \to \prod_X (-_x)$ is mono for every set $X$, and $C_3$ if it is cocomplete with exact filtered colimits. 

In bicomplete abelian categories the property in the OP is equivalent to $C_2$.

It's obvious that complete $C_3$ category is $C_2$, and $C_2$ always implies $C_1$ (it's in the 3rd chapter of Mitchell's book).

In the paper *Exactness of direct limits for abelian categories with an injective cogenerator* by L. Positselski and J. Stovicek [arXiv][1] one of the results is the following.

*Theorem*. Let $\mathrm A$ be a complete abelian category with an injective cogenerator $W$. (It is automatically cocomplete by special adjoint functor theorem.) Following conditions are equivalent: 

 - filtered colimits in $\mathrm A$ are exact;
 - $\eta_{W, X}: \coprod_X W \to \prod_X W$ is mono;
 - every morphism $f: W^{(X)} \to W$ factors through $\eta_{W, X}$;
 - summation morphism $\sigma_{W, X}: W^{(X)} \to W$ factors through $\eta_{W, X}$.

As far as I know, there's no known example without injective cogenerator, $\eta$ being mono and non-exact filtered colimits. If there are no products (and, necessarily, no injective cogenerator), I suspect that this "local finiteness" condition on coproducts behaves quite patologically; it seems very unlikely to imply exactness of filtered colimits. Construction of a cocomplete but not complete abelian category is already quite involved; I haven't checked (yet) whether 
[this][2] construction by J. Rickard serves as a counterexample to the updated question.


  [1]: https://arxiv.org/abs/1805.05156
  [2]: https://arxiv.org/abs/1805.10682