The main issue with this kind of regional identities, is that they develloped only after greater ones (Spanish, French, did). Without political structure to enforce it, their devellopement is pretty much limited by the presence of bigger that are overlapping them.
Furthermore, Euskadi as Occitania, or "linguistical" indentities have borders pretty much ignoring or avoiding geographical, regional, economical, sustainable borders. An independent Navarre would be much probably limited by OTL borders, more or less modified, that covered Castillan, Basque and Occitan (Gascon) places.
Regarding Navarre, they were pretty stuck by their neighbours without great hope of expansion southwards (to not stay north). It managed to grow mostly because its neighbours had more fructuous targets and because it served (from time to time) of buffer state between Castille, Aragon and France whom the de facto ownership was the cause of some civil wars.
Eventually, it didn't helped that Navarrese kings happened to have great estates in France, making them more french nobles in all regards than actual kings.
Finally, even if Navarre manages to remain some sort of bufferstate, like a nerfed Andorra or a Luxemburg at best, it would be certainly, as pointed by B_Munroe, a non-linguistic nationalism that would pop, an original Navarrese rather than Basque or Spanish.