Unfortunately, I do not have the 1938 trade directory for Birmingham, a bit of a loss in this case. I have the 1935 and 1936 directories in which Harry Hill is listed at the Villa Tavern.
The 1937 directory records Joseph Alfred Smith as licensee.
The 1939 directory has Horace Noel Smith as the licensee.
The 1940 directory, of course, shows Ernest Thomas Dickinson - he was granted the licence on July 25th 1939. This highlights the flexible nature of the directories.
Joseph Willis could possibly have appeared in the 1938 edition but, as I say, I do not have this directory. They may hold a copy in Birmingham library? If listed in the 1938 edition, it could mean he and Catherine were there in 1937 as there is some slippage between the survey and publication date. What I do know is that Joseph Smith left the pub in March 1937.
So, the possible dates for Joseph and Catherine being at the Villa Tavern can be narrowed down to post-March 1937 and the time of Horace Smith, who was possibly there in the second half of 1938.
Birmingham library should have an electoral roll for 1937-8, almost certainly their time at the Villa Tavern. Anybody going to the library soon?
Hope that helps.
Seasonal felicitations to all, Kieron