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We had the same situation on a recent cruise. We booked with CPS and it all worked very smoothly - handed the car keys over on arrival at the Ocean Terminal and on arrival back at the Mayflower Terminal we picked the keys up from CPS in the car park opposite the Terminal buildings and were away within minutes.
CPS Parking are the official Cunard parking partner. The car park for the Mayflower Terminal is adjacent to the Terminal. When you arrive at the Terminal the porters relieve you of your cases and a CPS agent will deal with your car and drive it to the appropiate car park, therefore no walking with you cases. In your case it will be driven to the City Terminal car park for you to collect on your return. Collect your luggage in the Baggage Hall, wheel your trolley out and pick up your car keys. There will be staff on duty to show you where to collect your keys.
Try Parking4cruises, very reasonable rates for a meet and greet service. They would take your car from you at Mayflower and bring it back to City. You should find them much cheaper than ACP.