I think the apps always tries to use available translations/transliterations for artists that is the language that is configured to be used for the UI.
In the mobile apps, I think it is always the same language as your system settings. But if the UI is not available in your system language, the UI will probably fallback to English.
In the desktop app, it seems you can set it in Preferences in the client only. The web player might use your browser settings. Since you are seeing transliterated artist names in the desktop clients, it must mean that your language setting is English there and something else on your Android device so it fallbacks to the default name, which might be cyrillic.
I just tested with: https://open.spotify.com/artist/3KYkCbCf8S3RDAtFoBT3qW
which is displayed as Valeriya if I have English in the settings and Валерия with all other languages.