OSM only provides tiles in the preferred local language of the area mapped, and only provides them as an aid to mapping.
You will need to check the usage policies of any other source of tiles.
The link you provide seems to use multiple languages, not just English. Germany is not even in English. And Green Park, in London is in German! I can’t actually work out what policy they use.
The cycle and transport maps seem to use English/Latin script, where possible, together with a non-Latin script, if appropriate. As these are loss leaders for a commercial map provider, I doubt they have a liberal policy on tile re-use.
I assume that anyone prepared to allow re-use of their tiles will tell you how they are named, although this should be easy to reverse engineer.
You also have to be careful with “download the tiles”. Most tile servers have some limitations. In general you are no supposed to scrape the server to download all tiles (i.e. in all zoom levels for the whole world).
Several OSM-based tile servers release their style as an open source project, so you can set up your own tile server with that style.
What is it that you want to achieve by downloading all tiles ?