Papazulu and relax.jet
The CAA FE quoted is correct as far as aircraft certificated for single pilot operations are concerned. As for UK airline operations it is normal for the F/O to log P1 (U/S) whilst "acting as pilot-in-command under the supervision of the pilot-in-command" (those words from the instructions for use in cap 407, the CAA pilots log book, by the way). Most take this as being on their handling sectors, but theoretically that isn't required, however they should carry out all the duties of the commander including all fuel/wx decisions and filling in tech log, checking load sheets etc.
There is no requirement for the PIC to be any sort of training captain. At other airlines I flew for we used to countersign F/Os log books to validate the entry but no one has asked me to do so for a long time. The commander still logs the flight as P1 as he/she remains legally in charge of the aircraft.