If you're paying NICs in the UK rather than cotisations in France, then your healthcare is funded by the NHS and not by France. You should therefore have obtained a workers S1 and used that to register with CPAM. CPAM will then give you a sécu number and a carte vitale, and will make the usual reimbursements which they in turn will then reclaim from the NHS. But without either an S1 from the UK because you're paying NICs, or French cotisations being paid by you and your employer, I don't see how you have any entitlement to cover from CPAM.
To put it simply - if you're paying your NICs in the UK instead of paying cotisations in France, you are the UK's responsibility and not France's. So France won't fund your healthcare. They will only pay out for you if they know they'll get the money back from the UK.
EDIT - just reread your post and seen your "application for a carte vitale is in the system" so perhaps I've misunderstood the entire situation. Do you mean you've submitted your S1 and are waiting for the carte vitale? if so did they not give you an Attestation de droits, pending the issue of the carte vitale? and if so, this will have your number on it. Obviously, if you're working for a UK company you can't apply for a carte vitale as a resident early retiree since you're not an early retiree...
What's confusing me I think, is that I don't understand what you mean by "registered as the employee of a UK company" - in what sense "registered", with whom? If the UK company has registered you as an employee with URSSAF, you should be receiving regular payslips showing your deductions, and your sécu number should be on those.