I've looked for this off and on over the years and haven't found anything I really liked. The closest I came to it was Youmail. It isn't an open source app, which is a bummer, but it seems to be a (at least sort of) privacy respecting company. I remember reading their privacy policy at one point and not seeing anything terrible in it.
Thank you, the 20 transcriptions allowed per month is more than enough for me. It is unfortunate that it's not open at all, but it is what it is, I suppose.
There isn't a turnkey solution I know of, but if you can have a tool save voicemails into a folder, you could probably have a script generate text files based on a Speech To Text (STT) solution:
Ahh, so this isn't a processing issue it's a data access issue.
Frankly, if you can't access the raw data of your voicemail inbox, probably no third party developer can too. This means that the only way to implement such a tool would to be to work with the voicemail provider. If they're a for-profit company, they probably have no incentive to make the data available unless there's a big moneybag involved somewhere in the exchange. That's probably why no such tool exists.
If you have a homelab or vps you could try rolling your own with https://hub.docker.com/r/onerahmet/openai-whisper-asr-webservice for the transcription and a foss Tasker alternative for retreiving the voicemail when your dialler app posts a voicemail notification. If you don't have the capability to selfhost whisper, you'd probably have to sacrifice the illusion of privacy (your mobile carrier will always have access to any calls made over their network) and use a transcription service from google/microsoft/openai.