How did the Muslim conquest end up strengthening Basque linguistic influence that was not there previously?
Quite simply, actually : Christian states surviving in northern highlands, and having their cores where Vascon or Cantabrian populations lived, you had an increased mutual influence.
With western Pyrenean regions being an important commercial and cultural road (would it be only trough Santiago pilgrimage) and with the North-South linguistical devellopment...
Interestingly, it could meant a more distinct Vascon or Basque language : more develloped or less so depends about the Gothic policies on it, and its capacity to survive politically.
Would more Gothic words and personal names survive without the Muslim conquest? Or were "barbarian" names like "Recceswinth", "Wamba" and "Wittiza" inevitably going to go out of style in favor of apostolic names (John-Juan, James-Joaquin, Joseph-Jose), kind of like how in England and France names like Offa, Egbert and Childebert eventually dropped out in favor George, Henry, John, Richard, William and Jean?
As for words...I'd expect the more romanized *Spanish, kind of like Occitan or Italian, with less germanic influence than French. Not to say you won't have, and doesn't have, a germanic-based vocabulary, but the absence of an Arabic superstrate isn't going to allow it to devellop it more.
As for names, tough...
You already have a cohabitation between Germanic and Romance names in Gothic Spain, so I don't see why it wouldn't continue.
The mentioned growth of evangelical, biblical and Christian names wasn't clearly established and was made trough "waves" : Carolingian era, re-evangelisation of the XIth century, XIVth century.
While I'd thing equivalents would most probably happen, for the middle term, germanic names are going to be present (while sharing partially with Roman-based names).
(I would point you, tough, that Richard and William are germanic names)
A question on the Franks- are they going to be a juggernaut in western Europe, no matter what, with or without the use of Poitiers & Tours to build up their reputations? Is the family of Charles Martel still destined for a royal or imperial future even without the Moors?
Assuming that the butterflies doesn't have major consequences on northern Gaul (and frankly, I wouldn't see why they would have), Charles Martel is still going to not onlt re-establish his father's hegemony in Francia but as well to increase it as IOTL.
As I tried to point
there (while it still a debatable topic), Peppinids were firmly in control of both Neustria and Austrasia, having put allies and clients in the latter after having crushed opponents.
It's less being "destined" than having made the necessary to clear the way before them and rooted possible opposition in the late VIIth and early VIIIth centuries (after the 710's, what we know about rebellions and inner conflicts involve Peppinids vs. Peppinids, and no longer external opponents)
For what matter to Frankish "juggernaut", with an early VIIIth PoD, it would be hardly surprizing to see them growing from a position that was already quite influential in Western Europe, to a really hegemonic one.
While the shift of dominant trade roads (for what mattered Francia and Western Europe) from Byzantine/Mediterranean continuum to Rhine/North Sea in the mid VIIth century, it made Northern Francia (and especially Austrasia, which wasn't exactly secondary before) and Frisians sitting over a commercial hub (giving the latter were firmly pagans, they soon loose their ability to sit on anything).
With an unified, wealthy, and demographically (therefore military) strong Francia, Peppinid hegemony isn't going to leave easily.
That said, ITTL, Aquitaine is certainly going to be stronger and wealthier, and its conquest is going to take more time (especially without the "double front" on Loire and Gothia). I'm not too sure Franks would have the motive to focus all their forces against it.
Interestingly all of this would motive Peppinids to go for Burgundy earlier than IOTL (would it be only because its kingship was tied with Neustrian) but as well to "compensate" for the partial unability to take on Aquitain wealth to fund their structuration (not only in a financial or material ressources point of view, but as an human ressource as well).
Eventually, tough, it would be enough of a necessity to have Frankish-Aquitain wars, as IOTL, focusing on Loire basin in a first time, which may be less fructuous with Eudon not distracted by his southern border.
Now, Eudon wasn't immortal, and waiting for his death and the succession conflicts among his sons could open opportunities : playing one against the other, effectivly dividing the kingdom/duchy in 4 or less (Vasconia, Auvergne, Gascony and Poitou).
We would be talking of less frenchified regions (It wouldn't surprise me if it would be helpful to maintain the linguistical border around Poitou longer than IOTL), with more strong identities and more prone, in case of crisis, to rebel.
IOTL, Charlemagne had to recreate a kingdom in Aquitaine to give room to these, while Franks had a clear domination (and partially because of the Arabo-Andalusian presence south of Pyrenees). Would ITTL Peppinids do the same?
I'd say they'd prefer to keep the region politically factured, and placing related families partially in charge (as with Guilhemids IOTL) would it be as a Gothic marche. Eventually, you'd have a (partial) replacement of local nobility, depending of their loyalty and rebelliousness, but I'd see more a mix (as you did have with Franco-Saxon nobility IOTL) than an outright replacement as IOTL.
Now domination of Aquitaine isn't a given, even in these circumstances. Things could play both in Frankish court and in Aquitaine to see the maintain of an unified entity south of Loire. Would this entity, tough, even with the Lombard alliance be able to maintain itself entierly?
I'm not sure, and while I admit there's still a possibility, I'd see more easily this unified Aquitaine keeping a sub-kingdom status (which would have really interesting consequences on Frankish structuration of defeated Christian duchies/kingdoms).
Could we see the maintain of Bavaria as a duchy within Francia? Would Benevent (already IOTL considered half a part of Francia during Charlemagne's reign) have a similar position (and therefore being considered as part of Francia ITTL if the conquest of Italy goes well).
Interestingly, it's this background that would look more as IOTL, with a parallel to make between the Carolingian Kingdom of Aquitaine, and this sub-kingdom of Aquitaine.
The obvious differences from IOTL would be the lack of clear prestige due to Tours. Giving that this prestige is essentially blostered up by Peppinid historiography, I've little doubt they would find something else to point out how much they rule, and how everybody else sucks.
The Pontifical/Frankish alliance was already in process, would it be only trough the increased Frankish presence in Germania (both in Frankish Germania and outer Germania) and a fight against pagans (Frisians, Saxons, etc.)
Eventually, it depends less from Franks than the butterflies of a more unified Visigothic Kingdom would have in Mediterranean basin.
An absence of Islamic conquest of Spain isn't going to stop Arab raids in Mediterranean sea : if something it may increase them in 740's against Italy, Spain and southern Gaul.
We're talking then about a Papacy that is not only threatened by Lombards but as well by Arabs, which would probably make rear-alliances necessary.
That said, could the Pope can play "the concurrence" with Goths and Franks? The latter are clearly more able, would it be only geographically, to deal with Lombards (which doesn't mean a conquest per say) and have a more strong political structure than Goths (which is why I think a partial "protectorate" of their kingdom by Franks may happen easily enough given the circumstances).
But, playing with the rivalties in the region may help the Papacy to gain a more important political and religious role than IOTL Carolingian Papacy (which isn't going to be hard).
Would the Franks still able to take over Lombards (would it be only because of the threat of rebellion it represent in Francia)? That depends.
I could see an hegemony being established (maybe earlier than IOTL, having
possibly Charles intervening in Italy), playing (as in Gothia) over political infighting.
Does that means full conquest? Franks would have the ressources for, but that doesn't mean it's a given that it would happen as IOTL : again more local alliances and autonomy
may be expected.
So, while Frankish hegemony would probably still arise, the main question is how this hegemony would be established and how much would it differ from IOTL Carolingia, especially in Mediterranean Europe (while I don't see major differences happening in Northern and Central Europe)