How to put music onto iPod nano with Apple music

You can't. Apple Music is a subscription service, requiring Internet. The iPod nano has no capability to connect to the Internet.

how do I put my own music onto my iPod even with Apple music?

You can put songs that you own on your iPod nano, while being subscribed to the Apple Music service. It's done the same way as before, by syncing your iPod nano with your iTunes library. But there are a some conditions...

Like before, the song files must be located on your computer's local storage. iTunes cannot sync song files that are only stored in your iCloud Music Library, which is provided by the Apple Music service and accessed (for streaming) by your computer's iTunes library and your iPod touch.

If you did not (when you started Apple Music) and do not (currently) also subscribe to the iTunes Match service, downloaded song files from your iCloud Music Library have DRM (copy protection) for songs that "match" songs on the Apple Music service. They are the same as song files downloaded directly from the Apple Music service. This DRM prevents those songs from syncing to your iPod, even if stored locally on your computer.

Songs from your iTunes library that do not match songs on the Apple Music service were uploaded to your iCloud Music Library. Downloads of those song do not have DRM, and can be synced to your iPod.

In summary, song files must be stored locally to sync to your iPod. Unless you intentionally deleted your locally stored song files after signing up for Apple Music, those existing song files are still stored locally. If song files are downloaded from your iCloud Music Library, the ones that have DRM cannot sync to your iPod. Please provide more details about your situation, to get more details about how the above applies to you.