I recieved this very helpful explanation along with some useful jargon from Matt.
Dealing with a person under a train (also known as a 'one under', or for the more delicate tanoy announcers 'passenger action') takes ages. First of all the person jumps/slips/is pushed/is chased/is crowded in front of a train. The train stops sharpish and the driver gets the current turned off, calls the emergency services and then puts bars down on the tracks to stop current being recharged.
Meanwhile the line controller or network operations centre will be calling the police (the Met and the BTP), the ambulance service, the fire brigade, the LU Emergency Response Unit, the Duty Manager Trains and the Duty Manager Stations. All of these people are required to attend.
Lots of resources start arriving at the nearest station and then have to get to the casualty. This is easy if the person jumped off the platform but can take a while if they jumped off a bridge two miles from a station and a mile from the nearest road access (in these cases there may also be a delay determining the exact location of the patient if the train has hit them and they have been thrown clear). None of the emergency services will go onto the track until it is confirmed that the current is off and that all train movements have been stopped (just because there's no juice doesn't mean that trains can't move -- some are battery operated and others can coast along for quite a distance after the current is turned off underneath them).
Once everyone's on scene the paramedics (of which there will be several, usually accompanied by HEMS) will assess the patient. If they are still alive (and on the Tube a lot of them are) then the paramedics will start working to keep them that way while the fire brigade and Emergency Response Unit try to cut them free. This can take a long time -- sometimes over an hour -- because trains weigh several hundred tonnes and have lots of sharp things underneath them that the patient could be trapped under/in/around.
Meanwhile, BTP officers will have arrived and will be trying to establish exactly what happened and why. 'He jumped under the train' won't do. They will want statements from the driver of the train involved, any other staff who saw the incident and any members of the public on the platforms. They will also view the CCTV if possible. Again this takes time and until the police are satisfied that there is nothing suspicious about the incident then the station will stay closed as a crime scene.
At some point the fire brigade will succeed in freeing the patient or, sadly, the patient will die. Either way they have to be removed from the train and this takes time. Once this has happened, if the person is still alive the doctors will work on them on the platform until they are satisfied that the patient is stable enough to be taken to hospital (either by ambulance or helicopter). Obviously there can be no question of re-opening the station while this work is going on.
Once the patient has been taken away it's down to the police to say if the station needs to be preserved as a crime scene. Although the majority of these incidents are suicides or attempted suicides, a few are accidents and a very small number are deliberate. You would not be happy if a member of your family was pushed under a train and the police didn't bother to investigate it but rather assumed that it was suicide so they could get the station open quicker. So until the incident is declared non-suspicious by the senior police officer on scene the station will stay closed.
If the patient dies at the station then there is a further delay because the coroner's officer has to be informed before the body can be moved. Some coroner's officers can be difficult to get in touch with (some offices close for lunch) so this may take some time. Once they have given permission then an undertaker has to attend to remove the body to the mortuary. Undertakers are commercial enterprises and so they will come when they can fit it into their rounds.
Once the body has been taken away or the patient removed to hospital, the police have concluded their investigation, all the services have packed up their equipment (which can take some time if they've just jacked up a whole train carriage) and any necessary cleaning has been done, then the station is ready to be re-opened.
In conclusion, everyone involved knows that a lot of passengers are being inconvenienced but they also know that someone is either dead or fighting for their life. If it was someone you knew then you would want it done properly and so we make sure it is done properly every time. If you get delayed or diverted then just think that you will be going home to your family tonight. Somewhere in the city someone's family are either being told by the police that their relative is dead or being rushed to hospital to see them while they're still alive.
Thanks for that Matt. So let that be a lesson to you selfish fuckers that consider jumping. Look at all that trouble you are causing. Take some haleys or shoot yourself for for sake of others. If not then bloody well cheers up and get on with it like the rest of us.