Police Community Support Officers (PCSOs) do not routinely work night shifts and would not be walking alone through London's East End at 2am. If the PCSO in question had been on a late shift it's unlikely a fire (even with an element of public disorder) would take that long to deal with.
PCSOs, like police officers, use fairly strict radio procedure. She would not say "PCSO Bousefield to Control" and they certainly wouldn't reply "Go ahead Mary". She would use a two letter station code (i.e. "Charlie Xray receiving 5714?" and they would reply "5714 go ahead".
Looking through paper files to find a victim with 39 stab wounds? Up until the 1990s, yes. Now there is a computerized crime report system and specialist software for murder / attempted murder which would allow police to find incidents very quickly.