Simple problem: Mr Vegas will speak normally, but as soon as a song is due to start, silence.

First I tough it'll be some copyright bullshit, but thinking about that, NPC and locations radio seems to work, it is only the pipboy radio which seems broken.

Read some posts talking about codec packs. I am on Linux, and codecs should be already installed, Can't use codecs from .exes anyway

Definitely sounds like a codec issue. I found a post where someone has solved this issue:
https://www.reddit.com/r/wine_gaming/comments/ghxgum/how_to_get_the_radio_working_in_fallout_new_vegas/fqcgom5

The original guide this guy mentions is gone, but with this comment's tips you might be able to solve your issue.

New Vegas runs flawlessly these days but still no music on the pip-boy radio. Even the main menu music disappears with Wine 6+

EDIT: Fixed it! Debian Wine 5.0 + Winetricks for installing the MP3 codec (l3codecx), LAV Filters (lavfilters) and OpenCodecs (ogg). All music is back.

You are the man! Spent hours trying to find info on this, it was bugging the hell out of me. Thanks so much for the tip.

This fix didn't work for me (Ubuntu, Wine 5). First of all there is no lavfilters component in winetricks for me, so I installed it from here manually (i.e. downloaded the installer and ran it from the command line with wine). Now when I play the radio in NV, my game just instantly crashes.