What would Europe have been like?
It's unlikely that Italy would be shattered among city-states at this point. Rome still had a large influence over its dominion even at the worst of the war and only some challenger cities saw the opportunity to raise as a regional power. Let's say Hannibal offer some agreeable terms to the Senate (which is only how they could be met, giving the Puno-Gallic army lacked means to besiege and takeover Rome) and make it a central italian power once more.
The planned league of southern Italian cities would probably come into fruition, it would be a whole mess : these cities couldn't yet begin to agree on anything, already began to fight each other, neighboured a lot of Roman colonies or pro-roman peoples, and Hannibal simply couldn't garrison his armies there or ask his Italian allies to supply them freely.
Contrary to Sicily that was a main strategic point for them, Carthaginians didn't really cared for Southern Italy safe in this case as a buffer against Rome. Which, due to its aformentioned structural and "original" dysfunction wouldn't serve this purpose much.
It doesn't mean that Rome would entierely recover and quickly so, and probably prevented to rise as a sole superpower, but as a regional power it will taking the form of a Tusco-Roman state IMO rather than Campano-Roman (even if I think that you'd have three parts eventually) possily more focused on taking back Cisalpina and Adriatic basin at first giving most of the outer threats (real or pretexted) would come for this.
Carthage on the other hand, would have to face two things : first a growing dichotomy between his usual hegemonic model (ties and treaties from cities to cities, rather than political or clientele) and a bloc tied to Carthage but essentially a familial heirloom of the Barcids in Spain and in negighbouring areas (notably thanks to mercenariship and personal relations). I think a mix between a Social War and a Civil War (roughly Barcids and their spanish autocracy against opponants in 'M, Carthage and allies) could quite possibly happen (with Romans intervening in sort of Punic War 2.5 to make their retun in southern Italy acknowledged)
Gaul, would probably remains roughly the same at first, but the renewed connection with Spain and continuity of exchanges with Italy might bolster just enough local archêi (Arvernoi notably, possibly Salues and Arecomicoi) trough trade and mercenariate (in Spain, or in northern Italy as IOTL) reinforcing their mediterranean tropes and stratification in the way, possibly erupting into quasi high-kingships defined by Gallic regions (themselves defined by assemblies and coalitions, as IOTL). At least, it's why I plan for my own TL, but I don't think being very far from plausibility there.
Rome might intervene in southern Gaul, would it be only to save Massallia's ass, but I would see this kind of intervention rather limited, possibly up to the Rhone river. I wouldn't see Punic influence being this much present in Celtica, except regarding trade goods.
On the other hand, you might see some Puno-Iberic or Puno-Celtic formations in Spain, depending how good Barcids or their successor do there.
Germanic Migrations say hello. While they may run out of steam before breaking into Iberia, I would still bet money on the assorted Goths on pushing into Gaul if I was forced to pick a side, and the factors driving the Germans weren't heavily impacted by Rome.
The main problem with your argument there is that Barbarian migrations were essentially tied not only to climatic changes, but as well to the existance of a sole political/cultural/economic superpower sitting in the Mediterranean basin. Goths as a people simply idn't emerged before mix of Dacians, Sarmatians, Romans, Celts and of course Germans mixed up in modern Romania as a coalition both targeted and subsided by the empire.
ITTL, the migrations of the IInd BCE century (Cimbri and Teutoni) might be a lot less unidirectional than IOTL, and more in dispersed order (some in Gaul, some in Italy, some in Balkans, why not some in Britain).
And giving that the clear differenciation between Germanic and Celtic people in Germania, and the final overcome of these, is directly attributable to Rome's presence (which did a lot to destructure Celto-Germanic peoples, even if mostly but not only undirectly).
You'd have migrations from time to time, but probably not in the same extent, not in the same ways, not in the same political frame which stabilized Migrating peoples (and eventually mixed them and accultured them even before their entered Romania).