It's the simple question of who (or which company) first manufactured and offered condenser microphones commercially--and the answer is Western Electric, several years before Neumann. They supplied condenser microphones to film studios by 1925-26, and Victor and Columbia signed up to use their equipment, including condenser microphones, in the switchover from acoustic to electrical recording of phonograph records. Some WE and RCA condenser microphones (the latter manufactured under a cross-licensing agreement) were used in radio broadcasting by the mid-1920s as well.
WE made condenser microphones well before those dates, e.g. for public address systems as early as 1921, but I'm not sure whether the microphones were available generally that far back. E.C. Wente's patent application for the basic design was filed in 1916. The workings of the capsule were also detailed in articles in The Physical Review starting in 1917. A T & T used these "condenser transmitters" for research and measurement within the telephone system (which is what they were invented for) for several years prior to their commercial introduction. The designs underwent several stages of development, such as aluminum-alloy membranes rather than stainless steel, and the use of grooves in the backplate to improve the damping of the membrane around the system's resonant frequency, between 1916 and 1921-22. By then the microphones had a wider frequency range than any recording or broadcast media that could convey their signals.
Unfortunately the publicly available information regarding types, specifications, and years of introduction is incomplete at best. A T & T's archives are private, and the people who write about old microphones, sound for film, the record industry, radio broadcasting and telephony of the era don't seem to talk to one another. The most important WE capsule types seem to be 370W, 361W and especially 394W (see attached page from a 1931 article in Bell Systems Technical Journal). Not many are in collectors' hands, since as a rule they were leased, then returned after the leases expired.