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

[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-20/issue-1/Representations-of-direct-products-of-finite-groups/pjm/1102992967.full)

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.