There are several different attempts to produce directories of the exit games that exist. This site has a directory of sites in the UK and Ireland, along with a timeline tracking the dates of openings, closures, moves and more. However, there are also larger-scale attempts to track the field, including a European map and global listings at Intervirals, Escape Rooms Directory, Escape Room Hub, Play Exit Games and Escape Reviewer. If you have a new exit game, you’ll want to submit the details to all of them.
The good news is that you can do so via the Escape Room Submitter which will submit the details to (at the time of writing) six listings at once. The listings are processed manually, so it may take some time to see the listings updated. You may use the site for a game in any country.