I was able to get reliable WAV file recording working, using an arduino nano, and a VS1053 circuit board (that also contained an SD card writer).
I wasn't able to find an off-the-shelf solution to this problem, so spent a couple of days working through some options before putting this pull request together:
It might not work for everybody (I'm not sure if it will work with those ESP32 boards) but it should be a real headstart for anyone trying to save .wav files directly from a VS1053 circuit.
The reason I wanted to do this, was that the default Adafruit code used Ogg Vorbis recording, which is a bit unusual, and had a bit of a delay before recording started.
The VS1053 chip is supposed to be able to generate the signal for a WAV file natively, so it should be possible to get it to save .WAV files instead of .ogg files.
I'm very happy to have solved this problem, I really hope this helps someone else.
If anyone wants to try saving WAV files without replacing the default Adafruit VS1053 library, I've posted the following sketch which works with the default libraries: