If you load a logo (I put a Waverley BC logo on Frensham Pond), and then from the preview and download page you select a different bathing water (note not controlled by the same authority) then the logo remains as per all your other options.
This is unlikely to happen in practice, as users should only be putting up signs for one local authority. It's also easy to change the option to remove the logo, if desired.
If you load a logo (I put a Waverley BC logo on Frensham Pond), and then from the preview and download page you select a different bathing water (note not controlled by the same authority) then the logo remains as per all your other options.