The logic is that the Pieds-Noirs lose a great deal of their political clout once the central government relocates to Algeria. They can no longer rule the place like petty colonial overlords with the cabinet, the army and the metropolitan elites looking over their shoulder. Since several members of the cabinet are the same people who first tried to implement the aforementioned reforms in 1936 (including Blum himself), there's a sense that "this time we're getting it done no matter what." There's a war on, what are local opponents going to do, take up arms against the government? Even then it takes until 1942 to finally abolish the Code de l'Indigénat and the practice of forced labor.
It wouldn't have been possible in peacetime (and indeed it wasn't) but with a reform-minded government acting under wartime powers, the situation is different. The kind of opposition that would have been crippling can be overruled, and the government deliberately starts building a support base among the Muslim majority by extending the franchise and revoking discriminatory legislation, so that eventually the Pieds-Noirs will simply be outvoted.
Whether that's plausible is up to debate, but that's what the authors went with, and I for one like the idea.