The First Punic War forms something of a gap in our knowledge. About three books of Livy dealt with it, but they are lost, as are the books dealing with it from Dionysus of Halicarnassus and Dio Cassius. We are largely reliant on Polybius, who deals with the First Punic war as a prologue and isn;t really interested in Sardinia anyways, so our knowledge of the circumstances of Rome's conquest of Corsica and sardina basically comes down to these two passages (Polybius I.79, 87)

About the same time the mercenaries who garrisoned Sardinia, emulous of the exploits of Mathos and Spendius, attacked the Carthaginians in the island. They began by shutting up in the citadel and putting to death Bostar, the commander of the foreign contingent, and his compatriots. Next, when the Carthaginians sent Hanno over in command of a fresh force, this force deserted him and joined the mutineers, who thereupon took him prisoner and at once crucified him. After this, devising the most exquisite torments, they tortured and murdered all the Carthaginians in the island, and when they had got all the towns into their power continued to hold forcible possession of Sardinia, until they quarrelled with the natives, and were driven out by them to Italy. Thus was Sardinia lost to the Carthaginians, an island of great extent, most thickly populated and most fertile. Most authors have described it at length, and I do not think it necessary to repeat statements which no one disputes.

The Romans about the same time, on the invitation of the mercenaries who had deserted to them from Sardinia, undertook an expedition to that island. When the Carthaginians objected on the ground that the sovereignty of Sardinia was rather their own than Rome's, and began preparations for punishing those who were the cause of its revolt, the Romans made this the pretext of declaring war on them, alleging that the preparations were not against Sardinia, but against themselves. The Carthaginians, who had barely escaped destruction in this last war, were in every respect ill-fitted at this moment to resume hostilities with Rome. Yielding therefore to circumstances, they not only gave up Sardinia, but agreed to pay a further sum of twelve hundred talents to the Romans to avoid going to war for the present. Such then was the nature of these events.

The epitome (basically paragraph long summary) of Livy Book XX also says this:

[Y.R. 512. B.C. 240.] An army sent against the Ligurians; being the first war with that state. The Sardinians and Corsicans rebel, and are subdued.

It is difficult to know whether this rebellion was the one described by Polybius against the Carthaginians, or one against the invading Roman forces (Polybius Book II briefly mentions the consul Gaius Atilius had set off to Sardinia).

This isn't much to build on, but there is an important takeaway, that is that Sardinia rebelled against Carthage pretty much as soon as the war was over and had effected its own independence. This was seemingly a two stage process: first, mercenaries who had been stranded on the islands took over the cities on the coast, then were overthrown by native Sardinians from the interior. This division between coast and interior is, I think, crucial: there is good reason to see Carthaginian Sardinia as being essentially confined to the coast and a few lowland areas inland. The interior was still inhabited by the builders of the nuraghe, and while I think they had stopped being built a few centuries before they were still inhabited.

This situation continued into the Roman period, and the Sardinaian highlands were considered a barbarian area (where it gets its name, Barbagia) by Cicero and Strabo. Strabo describes the ongoing military occupation (unusual for an interior province at this time) and continued conflict with the peropld of the interior (V.2.7--this begins with a brief description of Corsica, and it seems the same applies there):

There are four tribes of the mountaineers, the Parati, the Sossinati, the Balari, and the Aconites, and they live in caverns; but if they do hold a bit of land that is fit for sowing, they do not sow even this diligently; instead, they pillage the lands of the farmers — not only of the farmers on the island, but they actually sail against the people on the opposite coast, the Pisatae in particular. Now the military governors who are sent to the island resist the mountaineers part of the time, but sometimes they grow weary of it — when it is not profitable continuously to maintain a camp in unhealthful places, and then the only thing left for them is to employ stratagems; and so, having observed a certain custom of the barbarians (who come together after their forays for a general celebration extending over several days), attack them at that time and overpower many of them.

So even though the Romans were generally more succesful at controlling Sardina (seen by the replacement of early language with Sardinian) and the lowland and coastal regions were important agricultural exporters to Rome, the highland interior aggressively maintained its independence.