Basically everything stops as you need to get the BTP and ambulance down there and make sure they pick up all the bits.
Depending on circumstances it may be necessary for the TOC to find coaches to get people round stations but overall its about a 2 delay tops depending on where it happened.
From a signallers point of view - They will probably receive an Emergency call from the driver via the CSR if available. They need to find out more details such as the location & affected lines. They will then attempt to stop all trains in/heading towards the affected area by sending CSR Train Stop messages. At some stage they will contact the affected drivers and tell them the reason for the train stop message/change of aspect. They would replace the route on a NX Panel & throw the levers back in the frame in a lever frame box. If another signal box is involved they will send obstruction danger (six bells) to the other boxes in an absolute block area, and send the emergency alarm to a TCB/other method signalbox. Where possible they will stop the trains in platforms as they may be stopped for some time. They will then ring for an ambulance as of course the person could still be alive.
If the line(s) are electrified they will arrange for an Emergency Isolation with the ECR who will give a unique reference number that needs to be quoted if the isolation is been given up. If the signaller goes off duty they inform the ECR that they are handing the permit over the ECR will speak to the relieving signaller to ensure that they have a full and complete understanding. The signaller will report the matter to the NWR Route Control who will arrange for any diversions by speaking to the neccessary signalboxes.
They then speak to the TOC Control who will arrange for alternative transport, check route knowledge of driver/guards on diverted services, will arrange for a manager to attend the site to look after the driver (chain of care) & a whole raft of other things. They will send pager messages out to the staff who can then act in a customer care capacity with the ensuing delays.
The MOM will be called out & will probably act as the Rail Incident Officer The signaller tends to patch the driver through to BTP to report what happened the driver will then say if the saw anything Suspicious. Sometimes Route Control reports the matter to BTP & arranges for the driver to be interviewed. If unlucky the local police turn up as BTP may be some distance away. I remember one time the locals turned up & straight away treated it as a scene of crime. Which meant they were preventing the MOM getting onto the tracks. NWR had to then speak to the Police Control. In some cases the locals have headed straight down onto the track without checking the lines were blocked/isolated.
The locomotive/unit will also have to be examined & tested in regards the Brakes,Speedo,Horn & headlight. The train cannot be cleaned until the Police have given permission.
NWR Control will call a doctor to attend site to pronounce the death.
NWR Control will then contact the on call undertakers usually the Co-Op, to arrange for the removal of the body/remains. The body cannot be moved or covered until the Police give permission. The Police & MOM usually have to comb the area looking for body parts as obviously they can be found some considerable distance from the initial impact. Sometimes when there is an adjacant line that is not affected they may want to run a train past the affected area, the body is covered and not too visible to the passengers but they would have to speak to the driver to see if they would be happy to drive past the body.
As a previous poster stated they ideally want to have the line fully reopened within two hours, otherwise there has to be an Investigation by NWR & the TOC.
The RSSB Rule Book Module that covers this is M5 Managing Accidents Section 4 Bodies on the Line.