The paper Representations of direct products of finite groups. Burton Fein Source: Pacific J. Math. Volume 20, Number 1 (1967), 45-58.

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102992967

has what you are looking for and also explains what happens in the case that $K$ is not a splitting field. Look at Theorem 2.2 and the remark following it.

The paper Representations of direct products of finite groups. Burton Fein Source: Pacific J. Math. Volume 20, Number 1 (1967), 45-58.

Link

has what you are looking for and also explains what happens in the case that $K$ is not a splitting field. Look at Theorem 2.2 and the remark following it.