The process, assuming a fatality, is:
Incident happens
Driver stops train
Driver advises siggy and guard if there is one
Siggy advises control, and in third rail areas, Electrical Control to discharge the current.
Control advise BTP, MOM, TOC and National Control
Control declares major incident and sets up Command Structure (GOLD SILVER BRONZE(S))
MOM and BTP response officers deployed from wherever they happen to be.
If there is any suggestion that the individual survived, paramedics deployed
National Control advise undertakers who deploy from their base.
TOC deploy a replacement driver and / or Driver Manager
Driver, if in a reasonable state of mind, is patched through to BTP Duty Inspector via GSM-R to give a statement that advises circumstances of incident.
BTP duty inspector declares incident based on drivers statement - non suspicious, unexplained or suspicious.
If Driver unable to give statement, or the incident is not 'non suspicious', BTP will seek other witnesses and evidence on site to determine further.
MOM and BTP arrive on site, and conduct initial assessment, reporting back to their respective controls. Further resources, eg fitters for the train, or specialist cleaners may be requested, which control then deploy.
If non-suspicious, recovery of body can commence. If body is also out of public view, including from passing trains, unaffected lines can be reopened (and third rail recharged), once all personnel are confirmed clear of those lines.
If the incident is not declared 'non suspicious' after initial BTP investigations are complete, the body must be forensically recovered by Scenes of Crimes Officers (SOCO). They must be called to site by the BTP.
Body recovered, and either removed from railway property (by paramedics if involved, sometimes by BTP or MOMs, although neither are obliged to) or screened from the public for later recovery when undertakers arrive.
At this point, unaffected lines can reopen (and third rail recharged) once all personnel are confirmed clear of the track.
If train undamaged, it can be moved once the replacement driver arrives, or if the incident driver feels up to it (those that do have balls of steel in my opinion, and that includes the fairer sex).
Train will proceed to next suitable station to detrain passengers and be taken out of service, and sent to nearest depot for a full check by fitters.
Area of incident checked for any sign of incident. If all ok, line (and station, if affected) reopened. If not, the line or station must remain closed until specialist cleaners have done their work.
Put estimated times against all of that; some will have significant variation depending on incident location and the location of the nearest MOM, BTP response officers, and replacement driver. If SOCO are involved it adds at least 2 hours. If fitters are required, at least an extra hour.
In my experience, approximately, 60% are immediately declared non-suspicious, 30% are initially declared unexplained but subsequently declared non-suspicious following further site investigation before the incident concludes, 9% remain unexplained until the incident concludes, and fewer than 1% are suspicious.