(2-part post):
I used to run a large railway station. There are about 350 suicides on the railway each year, which means that roughly one line willed will be closed, for a short while, each day.
As Mushroom25 states, Network Rail can't re-open the line until permission has been obtained from British Transport Police. BTP work to a target of having lines fully operational within 100 minutes of an incident being reported. Given that BTP often only have a handful of officers on duty for an area of several hundred square miles, that's quite a hard target to meet. (Officers have to travel up to 50 miles, or more. They then have to carry out initial investigations to check that there are no suspicious circumstances before 'clearing up the mess' - which is sometimes spread over a mile of track - and ensuring that it's taken by an undertaker). Despite this, they often beat the 100 minute target although, on some occasions, it can take several hours to complete their work.
Additionally, the train operator has to send a relief driver (often by taxi, from a depot many miles distant) to take over from the driver involved in the incident (who is always automatically relieved from duty). In some circumstances, it's necessary to send a senior mechanic as well, to check that the train is safe to drive.