This is GPO Plug 420 - the standard for UK telephones from the mid-60s to the 80s when the post office spun off the phone system to British Telecom and switched to a plastic IDC plug. It's about 1/4" across, so it's nice and chunky. The jacks are 60s-quality engineering too. Hopefully the gears can rotated fairly freely and the jack will maintain contact - the 4 pins will be ground, power, audio out, and possibly a separate audio ground, if it needs it. The back of the baseboard will be a point-to-point wiring mess, but basically a passive mixer for all the grid of jacks, feeding into an amplified output. By using the last ring for power, they will only power up when fully clicked home.

I found a couple of surplus suppliers for these plugs at not-terrible prices yesterday. Considering there must have been millions in use at one point, they're surprisingly hard to find.