Vladimir Arnold is known, among other things, for offering a scathing critique of Bourbaki:
Recently I've been reading some Nietzsche, and he chides some Germans in the wake of the Franco-Prussian War for triumphantly proclaiming that the English and French might have civilization, but we have culture.
Inspired by reading this, I'm wondering if there are any critiques by 20th century German mathematicians of Bourbaki out there?
It seems to me that German-(speaking) mathematics at the advent of Bourbaki was primed to be, by and large, comfortable with trends towards axiomatization, abstraction, and structure theory in the footsteps of Hilbert and with the successes of abstract algebra in the Goettingen school around Emmy Noether and van der Waerden. See Alten et al "4000 Jahre Algebra" Springer, 2003 (in German).
There was some engagement around the question of reforming pre-university eduation in the late 60s and 70s. In his 1965 book "Mathematik als Bildungsgrundlage" (Mathematics as foundation of education"), Meschkowski dedicates a chapter titled "Bourbaki in der Schule?" (Bourbaki in schools?) to the question of ideas of Bourbaki (here primarily the advocacy of Dieudonne) making it into highschool education. While there is some critique (particularly the idea of replacing Euclidean geometry with vector spaces), it is nuanced and overall quite positive.
Carl Ludwig Siegel has been mentioned. As an addition to what has been said, I'd warn against reading his rejection of abstraction in algebra as a reaction against French mathematics. Siegel's 1959 letter to Weil give more context:
It is entirely clear to me what circumstances have led to the inexorable decline of mathematics from a very high level, within about 100 years, to its present nadir. The evil began with the ideas of Riemann, Dedekind and Cantor, through which the well-grounded spirit of Euler, Lagrange and Gauss was slowly eroded. Next the textbooks in the style of Hasse, Schreier, and van der Waerden, had further a detrimental effect upon the next generation of scholars. And finally the works of Bourbaki here provided the last fatal shove.
Notice how the mentioned drivers of Siegel's perceived "decline", namely Hasse, Schreier and van der Waerden as well as Riemann, Dedekind and Cantor are all German or German-speaking. (See Grauert's "Wie Gauß die alte Göttinger Mathematik schuf" in "Proceedings of the 2nd Gauss Symposium, 1993". Relevant quote is also cited in these more accessible German and English translated sources.) Grauert incidentally, citing this very letter to Weil, advocates for a nuanced view of Siegel in the letter to the editor linked by ThiKu. I encourage reading the letter (which isn't that long) in full.
This seems to be too long for a comment.
Siegel's emotional letter to Mordell concerning Lang's book "Diophantine geometry" compares Bourbaki style mathematicians (Siegel does not say "Bourbaki") with pig in a garden and even with national socialists:
"Thank you for the copy of your review of Lang's book. When I first saw this book, about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!
The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory — Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.
Unfortunately there are many "fellow-travelers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!"$^\ast$
I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up. Let us hope that your review may be helpful..."
The comments of Serge Lang were published in AMS Notices (42:3, 1995), see here from p.339
$^\ast)$ "We will continue to march, until everything shatters", from this popular Nazi song.
Another scathing critic of Bourbaki can be found in the article "The Ignorance of Bourbaki" by the Logician A.R.D. Mathias in THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, pp.4-13. So it is not a German-French argument, but nevertheless a harsh criticism of Bourbaki's stance on Logic and Meta-Mathematics.
Quoth R. Haag (2010, p. 280):
K. O. Friedrichs once took me to the home of Richard Courant to introduce me to his former teacher. I remember vividly one remark by the old master because I was at first thoroughly shocked and thought much about it. So I can still remember it almost verbatim: “There is a fascist group of mathematicians in Paris who have not understood that mathematical reasoning is a natural activity of the human mind.” Among my mathematical friends the work by the group which published under the pseudonym Bourbaki was very highly regarded and I was full of admiration for this unselfish collective effort at organizing all mathematical knowledge in logical order and elegant economical formulation. Why on earth did Courant classify this as fascist?
(One can find toned down versions of this — not naming Bourbaki — in Courant et al. (1962) or his NYU colleague Morris Kline’s “tirades” against the “modernists”.)