As Bill says, find your incoming stopcock and check that it's fully open. Do you have a separate supply from the downstairs flat? If you have your own water bill, than it must be separate.
Next, close the stopcock and see if that stops the water in the bathroom taps. If not, then they are fed from a tank (probably in your roof space.) It's not done now, but was quite common in the 70s.
If they're tank fed, there may be a long standing airlock in the system.