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Generally, individual branches of the armed forces were responsible for countering the codes of their allied opposition. The German naval intelligence agency (Beobachtungsdienst) broke the British naval codes.
The unified UK code-breaking effort was helped by a single system being used so widely across the German forces.
One of the ironic bits of cryptography history is that every side broke almost all the codes of their enemy while refusing to believe that their own codes were anything less than perfect.