(2-part post):
I've not worked on the London Underground but I used to run a (mainline) rail station. I suspect that the problems and procedures are fairly similar.
Approx 350 people a year (i.e. roughly one a day) commit suicide on the national rail network. During the 4 years I ran a medium-sized station, I never witnessed anyone actually 'jumping' but there was one occasion when I had to get all trains through the station brought to a halt because of a guy who was determined to end his life. However, there were plenty of occasions when trains were delayed because of a report of 'one under' at stations further along the line.
As soon as a 'one under' report is received, all trains in the area are brought to a halt, so that the situation can be assessed and the appropriate action taken. Not all suicide attempts are successful so, in some cases, emergency workers will have to try to get the person out from under the train, while giving the same sort of medical care that's given to people trapped in their cars after a road crash. In other cases, there's nothing to do other than to (quite literally) pick up the pieces.
On the Underground, the emergency teams can't do their jobs with the power switched on, so the section of track will be isolated while they're at work. To complicate matters further, there's always the possibility that the person was pushed (or that an employee was criminally negligent by allowing a dangerously large number of people onto the platform). For this reason, every incident is initially treated, by the British Transport Police, as a crime scene.