Hi HMF
The scenario you pose is not particularly complicated and it's quite common, not only in situations similar to that you describe but many others where two "projects" interface and it's certainly not new!
As example the first guidance on CDM Client duties published by the Construction Industry Research and Information Association (CIRIA Report 172) in 1998 specifically identified the common scenario of infrastructure developments that necessitate utilities works that are procured by others - in such a situation much of the utility works would be off the main site but with a need for coordination when projects interface.
Up to the Client to decide whether these additional works involve "construction work", thence such that CDM applies, and then to what extent CDM would apply to this different "project" [with both terms in quotes as defined in CDM].
Let's assume for a minute that the Client's different project means that CDM applies in full, though this doesn't actually matter much as you need to have regard to legislation whose roots are superior to CDM, ie the Management Regulations - see below.
They need to appoint a PD and PC, and the duties of Client, PD, PC, Designers and Contractors apply in exactly the same way as for the "project" with which you are involved.
The Client could then have two PCs working on the same site simultaneously or decide that one or other takes the lead.
If they want the PC for your project to do that, then the PC might reasonably want to be paid for something they have not (presumably) priced for - that's an issue for the Contract, not one of criminal law.
If they want the PC for these new works (which could of course be the Client themselves) to take the lead then they can still expect the PC for the main works to ask for more money to cater for the inconvenience etc.
Nothing in CDM says that you cannot have more than one of anything whether that be "project", "PC" etc.
If the Client decides not to make one or other PC the lead then they can expect conflict.
But this is dealt with by the provisions of the Management of Health and Safety at Work Regulations 1999 which requires cooperation between all the relevant parties.
In simple terms, that means making sure that everyone is aware of who is in charge of what and when. The Client could example determine that these new works should be done in a site within a site. So that the PC for the new works would be in charge of that area for the time and to the extent needed for them to do their work. If the PC for the main works needed to send people in, they would have to abide by the other PC's reasonable rules and directions.
Just needs people to sit down and work out the details of how this would work.
Good luck, Peter