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Nothing in either feat has any restrictions, so you can absolutely put these on the same character.
How to think of it: All gnomes have a connection to the first world, since Gnomes originated there a long time ago. In the interim, some gnomes have developed a connection to something else as well. Having both the wellspring heritage and the first world adept feat means that your connection to the first world is strong and that you've developed a connection to something else.
I interpret as First World Adept's requirement shifts to match your new tradition, which seems the intention of Wellspring Gnome: that your PC's Gnome magic gets tied to the new tradition, including the wording of other Gnome Feats. Otherwise there's not much point to that last line EXCEPT to specifically cut you off from First World Adept even if you take First World Magic. That'd be an odd intent.
So most magical lineages are fairly even for a Gnome, except you can swap the Primal Cantrip out every day & get the Fey trait so that's still the Gnome-iest one.
By legalistic RAW, you can't take it.
Then Wellspring only works on First World Magic.
I see nothing that would block access. All you need is a primal inate spell. For instance, a gnome rogue can pick up the minor magic feat at 2nd and pick 2 primal innate cantrips. First World Adept doesn't require that the innate spell comes from a gnome ancestry feat.