France was home to some of the greatest mathematicians of the 20th century such as Weil, Serre, Grothendieck, and Deligne, just to name a few. What algebra texts did they study from during their undergraduate and graduate years?
The mathematicians you name all had important relations with Bourbaki, Deligne for sure studied the elements of Bourbaki at that level. While the others would create them. The algebra of Bourbaki has a foundational status in the elements. But this is not the answer to your question, before Bourbaki (and shaping the way he will present their elements) it is Moderne Algebra of Van der Waerden.
He "presented ideas that had been developed earlier by Emmy Noether and Artin —whose courses van der Waerden had recently attended in Göttingen and Hamburg, respectively— and also by other algebraists, such as Ernst Steinitz, whose works van der Waerden had also studied under their guidance. Van der Waerden masterly incorporated a great deal of the important innovations accumulated over the last years at the level of the body of algebraic knowledge. But the originality and importance of his contribution is best recognized by focusing on the totally new way of conceiving the discipline it put forward. Van der Waerden presented systematically those mathematical branches then related with algebra, deriving all the relevant results from a single, unified perspective, and using similar concepts and methods for all those branches."[¹]
"It is true that there were already excellent monographs at the time and, in fact, Bourbaki treatise was modeled in the beginning on the excellent algebra treatise of Van der Waerden. [...] Since then, algebra has developed considerably, partly because of Van der Waerden's treatise, which is still an excellent introduction. I am often asked for advice on how to start out studying algebra, and to most people I say: First read Van der Waerden, in spite of what has been done since." [²]
You should see:
[¹] Mathematical Structures from Hilbert to Bourbaki by Leo Corry
[²] The Work of Nicholas Bourbaki by Jean A. Dieudonné