I'm guessing broken rails like this get reported by train crews, unless there's some way of detecting them as you'd think the track circuit would still be fine
Where there are track circuits, if the rail breaks in plain line and that rail is not a traction return rail, in many cases, the track circuit will show occupied with no train present.
In areas of point work or for a traction return rail, the track circuit may not be affected by the break as there may be an alternative current path.
Note that some types of track circuit may "jump" a break if the gap is small enough. Although this does not normally apply to DC types.
As far as the railway is concerned, track circuits are not designed to detect broken rails and hence there is no reliance on them doing so.
If a driver sees a broken rail (and at line speed this is far from easy) or notices a rough ride / bump, they will report it to the signaller. The signaller will then stop the next train and advise the driver of the report and ask the driver to proceed at caution.They will ask the driver to inspect the line and report back after passing through the signal section.
Meanwhile, the signaller will contact control to report the problem. Fault control will in turn arrange for an appropriate infrastructure maintenance team to attend. Control may also ask a MOM to attend.
If the break is found to meet a certain criteria, it may be possible for trains to run over the break at 5MPH. The Permanent Way track staff may be able to fly clamp it (fishplates held in place with G-clamps) in which case, the speed may be raised to 20MPH.
A permanent repair will need the line to be blocked. It may be possible to do this during the day. But it depends on the type of rail and the type of break, the condition of the sleepers as well as where it is. Otherwise it may have to wait until a T3 engineering occupation can be arranged at night or at a weekend.