Complex sentence: He made early stage appearances before infantry service in the First World War, after which he had his first major theatre success starring in "Kissing Time" when the musical transferred to the West End from Broadway in 1919.
Split into simpler sentences: He made early stage appearances before infantry service in the First World War. After the war he had his first major theatre success in "Kissing Time", which he starred in when the musical transferred to the West End from Broadway in 1919.