I believe engineering is a solid background for audio-dsp, because many fields meet: physics, acoustics, signals, mathematics, electronics, programming. I obtained a master degree in electronic engineering (long time ago), but have no further formal education in audio or DSP.
If you have the freedom, choose an audio-related topic for your master thesis? I worked on realtime polyphonic pitch recognition (in 1999). This is a very difficult problem, since the harmonics overlap heavily, and music signals can be really wild. When playing a single string on guitar, there is fret noise, the other strings resonating, the part of the string between the nut and the fingered fret resonating, body resonance, possibly palm muting, and subtle pitch bending, and that's only a single string. Even when I did not develop a commercially viable solution for real-time polyphonic pitch recognition, I developed an imperfect but functional real-time demo using multirate filterbank analysis and some heuristics. I could show the jury that an apparently simple problem is really not so simple, and a solution requires balancing trade-offs.
I believe Axoloti can be a nice tool to develop a master thesis. Many objects and techniques can be added to the library, that can align very well with a master thesis topic in electronic engineering.
I don't have a clear view on available courses, or clear suggestion how/where to learn more. So I'd love to hear others' recommendations here!