Okay, I was hoping to do explain this a bit at a time, as it's quite complex. Hence concentrating initially on the iPod. But never mind.
I was asking if you can find the songs listed on the iPod, either under the songs or the albums menu? Look in both menus. The iPod nano must have a method to display songs and albums. Every iPod does (with the obvious exception of the Shuffle) and that is independent of whether you used iTunes or the Music app to get the music onto the iPod. If you cannot see either of those menus, presumably they have been deselected in the iPod's Settings menu. If so, perhaps you can turn them on temporarily in order to check for the missing artists' songs.
If the songs are listed on the iPod in either the songs or albums menu, but not under that artist, that would suggest that there's an issue with the artist name field (tag). This is probably due to one or more of the factors mentioned below.
Regarding the field (tag) album is a compilation of songs by various artists:
on your computer, what needs to happen in the "album is a compilation of songs by various artists" field is;
if the album is by one artist (i.e. a solo artist or a single band):
- the compilation field must not be ticked (checked) on any track
and
- in each track on that album, the Album Artist field must either be blank* or have the name of the principal artist in it (* but read on below as there is more to this than one might think)
If either of the type of albums mentioned above (solo artist or single band) has guest artists on any track (i.e. the track is credited to the main artist featuring a <guest artist>), you must complete the Album Artist field with the name of the main artist only. When an album has guest artists listed in the Artist field, you should not leave the Album Artist field blank. Make sure that you identify both the Artist and the Album Artist fields. They do different things and failing to get these fields correctly filled in has different effects on different iPods. On my iPod Classic (and I think on my iPod Touch), if those fields are incorrect, it can place the album under the various artists section and not list it under the artist.
Obviously, guest artists will not be listed separately.
Next; compilation (Various Artists) albums:
Often, an album that is a collection of a single artist's hits from their career is listed on the (third-party) database used by Apple, as a compilation album. However, iTunes (and I therefore assume) the Music app, do not regard such albums as "compilations". Apple assume a compilation album to be a collection of songs by different artists, that is, a Various Artists album.
So if a single artist album (even their Greatest Hits albums) is marked as a compilation, the iPod will place it in the wrong menu. To emphasise, a single-artist album is not a compilation as far as Apple is concerned.
Lastly (and this is an important one):
if an artist only appears on a Various Artists album (that is, a true compilation album), which is correctly marked as a compilation album and with the text Various Artists in the Album Artist field, then that artist will not be listed under the artists menu on the iPod.
Note though, that if and when you add a single or album by that artist, not only will that artist appear in the artists menu, but any song on a compilation album will be listed on that menu as well.