I would guess the first station to use mp3 on a broadcast did it in the 1990s, it would have taken a computer connected to a transmitter then. I would think the big switch to digital audio would have happened when the when recordable CDs, and then portable mp3 players came out, allowing stations to broadcast remotely.
Here I would get mp3s online, then record them to tape, having nothing else but the one computer to play them at the time, with not enough space to save a whole collection of files.
Mp3 audio quality is not necessarily poor, I'm sure that's up for debate in audiophile circles, and analog sources can be recorded at all different qualities on tape. Its just that mp3 didn't set out to be a poor quality format. There were a few poor encoders early on though, and a bigger problem was that to save space on storage media, people stuck with encoding at lower bitrates, perhaps not noticing the quality because of being dazzled by the new tech.
Here in the future, if you record using high bitrates like 320k and with a better codec like Lame, subjective quality can be good.
Now we could be storing uncompressed files, but I'm glad for mp3 early on, it gave us audio we might not otherwise have. I'll continue to use it.