I recently started a new save, however, I can't edit the database on the top right-hand corner of the menu before clicking advanced setup. Can someone help me to solve this, as I want the updated version with VAR in the Premier League.
I'm assuming you're thinking there's a 19.2.1 database. There is not.
SI release two databases per FM; 19.1 and 19.3 for this FM. 19.1 comes out when FM is first released, 19.3 will come out sometime during March (probably).
The database is the data regarding players, staff & clubs.
League rules, etc, are handled by all the other updates. 19.2.1 isn't a database, hence why you cannot pick it when you start a new game.
To check if the 19.2.1 patch is the current one for your game, go to your Preferences page and you should be able to see it near the top of the page. If you have 19.2.1, starting a new game with 19.1 database will have all of the latest changes with regards to league rules, etc (including any changes made by SI for VAR).