How does France fight on? It's not as easy as it seems. IOTL, the war in Algeria was a hugely divisive issue in France, and it tied up a lot of political capital (not to mention economic capital), and it was already a war that exhausted France when it ended in 1962.
Without a POD before 1945 that fundamentally alters the demographic, political, and economic situation of France and Algeria, I don't really see it happening and working out for France.
The best (well, worse) situation for a continued fight that I can think of is this: some sort of quasi-fascist right-wing coup succeeds à la the OAS in the late 1950s or in 1961/62. This military junta (as I can imagine that only the military would be able to pull off such a coup) somehow manage to redouble French militarism and purge the pre-coup civilian government, plunging France back into Algeria with renewed vigour. This nightmarish scenario would result in a drastic escalation of the war as it evolves into a Vietnam-esque quagmire with the added dimension of an angry and scared pied-noir minority in the zones of conflict. Violence against the pied-noirs and resultant military reprisals become a circular system of ethnic cleavage. In such a situation, both white Frenchmen and Algerians become more and more radicalised, and more and more atrocities are committed by both sides. Even in this situation, France fundamentally lacks the economic and political wherewithal to subjugate millions of Algerians, and domestically, this hypothetical right-wing military junta has little civilian support in metropolitan France and no foreign allies save for (maybe!) Spain and Portugal. As the façade of benevolent colonialism (which IOTL was already very weak) drops completely, the international community reacts with disgust and horror. The idea of Algeria as an integral aspect of metropolitan France, and not a colony ruled through coercion and oppression, is exposed to everyone as the thin fiction that it was. The military junta eventually collapses either because of domestic/international antipathy or because it simply runs out of money, and the Algerian victory becomes a bloodbath as vengeful indigenous revolutionaries exact their revenge against the pied-noirs and the Harkis.
As we can see, the French are still defeated but what's more, they will lose in other spheres. Francophone Africa reacts with horror and expunges itself of the French neocolonial ties that turned out to be very profitable for France IOTL. The UK and the USA come to view France as an unreliable and myopic foreign policy partner. The legitimacy of a post-war democratic Europe is fundamentally undermined as Europe's third-largest country falls to militarism for the sake of a white settler minority. And white-minority settler colonies elsewhere in Africa (Rhodesia, South Africa, Angola, Namibia) either double-down and become even more draconian for fear of pied-noir massacres, or explode into ethnic violence as indigenous revolutionaries radicalise far faster than IOTL.
IOTL, France's successful decolonisation from Algeria, the successful evacuation and repatriation of pied-noirs with relatively little turmoil, and its ability to maintain close neocolonial ties despite the bloodshed that had occurred prior was a lynchpin moment in how countries viewed decolonisation in Africa. If you fuck with that, and you fuck with a lot of shit.