UA,
The FO will log the time as normal. P2 when non handling, PICUS when handling.
Before they can log any time, they need to have the correct type rating. If the type is single pilot, the chap(ette) passes his LST in a single pilot role (i.e. LHS). As the type is issued as a single pilot type, there is no requirement for an MCC to be completed. Chap then undergoes a RHS OPC and they are good to go on with line training.
Further to that, you do not need an ATPL to be Captain, only the minimums required to operate that aircraft in a Single Pilot Role (normally 700 hours).
Once you have 500 hours on a single pilot type operated as a multi pilot type, then you can apply to the CAA for an MCC exemption. All of the multi crew hours will also count towards your ATPL issue. However, the LST for an ATPL issue must be performed in a bona fide multi crew aircraft.
To operate a single pilot aircraft in a multi pilot role you normally need to involve the CAA (and pay the appropriate fee). Some aircraft types, although are listed as single pilot, cannot be operated on public transport flights legally without two crew. I think this applies to all pressurised aircraft (CJ1, King Air).
Clear? Thought not.....