Dave---
All Cunard White Star ships flew both flags from 1934 through the end of 1949. On the ships that originated with White Star, the White Star burgee was flown above the Cunard house flag; on those which originated with Cunard or (like QM) entered service after the merger, the Cunard flag was higher.
After 1 January 1950, even the remaining White Star ships (Britannic and Georgic) flew the Cunard flag above the White Star burgee. The one piece of this that I'm not sure of is what the rest of the Cunard fleet did from 1950 on. I believe, but have never confirmed, that as a general rule the rest of the fleet flew only the Cunard flag.