Jim,
First and foremost, do not fret. I had a similar situation on a CO award ticket for SQ flights were CO's SHARES and SQ's Kriscom simply stopped communicating with each other, such that SQ had no record of my booking. Made for an interesting experience at check-in at SIN. But, I made it onto the flight, and here's why you will make it on if you are polite but persistent:
An airline will honor a valid TICKET for one of its flights, even if there is no record of a *reservation* in the airline's computer system.
You have a valid ticket for Swiss flights. Therefore, Swiss must transport you on the flights and cabins booked. Print out your CO e-ticket receipts, and bring you Swiss confirmation code with you to the airport. When the agent does not see a reservation for you on the Swiss flights, give the agent your CO e-ticket numbers, as he or she can readily access the CO e-tickets, and will see there are open e-ticket coupons available for use for the specified Swiss flights. Since it sounds as if there's a live Swiss PNR for you, the Swiss check-in agent (or maybe the ticketing desk) can simply sell in First class seats - revenue seats - into your booking, and manually attach the CO ticket number.