I recently got the Nauticam A7SIII housing to shoot video and am very happy with it, shooting with the 28mm F2 and WWL-1 that I bought on this forum from @Phil Rudin
I just tested the Sony A1 in the same housing today, thoroughly checked all buttons, dials, etc and wanted to just share to anyone who needs to know, the A1 works perfectly in A7SIII housing.
I know for the Nauticam A1 housing they sell the adaptor tray for the A7S3, but this is not required if you have the A7S3 housing.