The system is a century old with original electromechanical controllers.
There's literally no excuse for not maintaining your infrastructure. Every probably can be solved. Shut down a line every night till the job is done. Or shut it down for a day or two. Is the line integrated with others? Maybe you'll have to shut down three or four lines. People will hate it. But if it avoids months and months of problems to have a few days, they'll cope.
What helps London is that the system shuts down for a few hours each night (except weekends on some lines). That allows most repairs and maintenance to take place without altering schedules.