Is there a table that governs spell names for different locales?

I would like to change the name and description of a custom spell for a different locale. For NPC names, we simply change creature_template_locale but there are no corresponding tables for spells.

How would I go about changing spell names & descriptions for different locales?

I looked through all available locales in acore_world and found nothing relevant.

Spells are read from DBC files. You can extract data from these files using e.g. Stoneharrys wowspelleditor and write the adjusted data into the spell_dbc table of the world database, which serves as an override.

Spell names/descriptions are contained in the client's DBC files, which you'd need a tool like Stoneharry's Spell Editor or WDBX Editor to edit. You'd also need a tool to open the MPQ files the DBCs are stored in, such as Ladik's MPQ Editor (both can be found on Github). You'd then need to repackage the DBC into a patch-#.MPQ file (or a folder named as patch-#.MPQ) and send it to any client you'd use to connect to the server, as well as the server itself. If you've already got your custom spell showing up in-game, you likely already know how this works.

Only including the updated locales in the spell_dbc table of your server as Honey55 suggested won't make the localized fields text appear for the clients that don't have the patch downloaded as far as I know.