As regards to callsigns used by the Military, most of the Squadron/Mission specific callsigns aren't approved by ICAO. Callsigns such as Ascot (RRR), RafAir (RFR), Rescue (SRG) & Army Air (AAC) are all allocated by ICAO with the relevant 3 letter code.
However, "Kestrel" for example is the callsign for MyTravel Airways (ICAO code MYT) but is also used by 20sqn Harriers
"Snoopy" is another one, It's the official callsign of a German Bizjet operator but is/was also used by Grob's of the Yorkshire AEF
There are plenty of other examples where Military units use callsigns which are allocated by the ICAO to Airlines etc.. I'm not sure who, if anybody, approves the use of callsigns by the Military (& it's not just in the UK)
Fried Chicken