Especially with the word "compass" in the title, I figured they made an accurate map of the earth's magnetic field and would use that to avoid needing any satellites. Then I opened the comment thread...

Come to think of it, the earth's magnetic pole shifts so I guess this isn't possible? Either way, "London Underground tests new dead reckoning system" would have been more accurate it sounds like

Regarding pole shift: you maybe could have multiple static measurement stations spread around which measure the current values at known positions and synchronize the data once per day to the trains. I’m not sure how you could even determine position just from magnetic field alone, surely there are multiple positions with the same magnetic conditions?

Maybe, I don't know enough about magnetic fields to say much. In my head they're directional so you'd know how it's aligned and can measure how strong it is, and that might function similar to a gravity map, especially if you know where you're coming from: there may be two places on earth with the same amount of gravity, or in this case, the same magnetic field strength, but if those places aren't next to each other that may still be tolerable depending on the system requirements.