For which finite groups $G$ does the Wedderburn decomposition of $\mathbb{Q}[G]$ consist only of fields and division algebras?

The groups you are looking for are precisely those for which the group algebra $\mathbb Q[G]$ does not contain nonzero nilpotent elements. These groups have been classified by Sehgal in here. The finite groups which have this property are the abelian ones and the Hamiltonian groups of order $2^mt$ (where $t$ is odd), such that $2$ has odd order modulo $t$.

The condition is clearly equivalent to require that the group algebra $\mathbb{Q}G$ has no nonzero nilpotent elements and there is already an answer here:

https://math.stackexchange.com/questions/906764/reduced-group-algebras