Telescope version 126.96.36.199 had a bug due to which popup fields would not show up in Advanced Search, while the popup fields would be active within the system because they did work in TSUploader & retrieving metadata information functions of Telescope.
This is further evidenced by the
tsp_getpopups stored procedure getting all the fields and values from the database as expected. This can further be tested by keeping only one field with popups, deleting all popups and have it active for just one field, or deleting all lookup configurations.
This issue was fixed in Telescope Executive 2020.1 and it is recommended to please upgrade your Telescope version to this or later to fix this issue. If your Telescope installation has customizations or you're an OnDemand customer, please contact your Account Manager or support for help with upgrading.
Once Telescope is upgraded, all the configured popup fields should be available in Advanced Search as expected.