CWLCY is a great suggestion, though the seats are in fact fully flat, lie flat generally being used to denote the semi-flat variants.
Here is some more info on the service, which is especially cost-effective if you purchase the ticket exAMS at around £2,250:
Club World business flights from London City to New York - A318 - British Airways
The Queen Mary 2 would also be a fun idea. Costs from about £2,500 one way each for their "business class" Princess Grille, and from £3,250 for "first class" Queen's Grille - it's great value when set against a business class airfare, all meals and accommodation for a week:
Queen Mary 2 Luxury Ocean Liner : Luxury Cruise Ships : Cunard Cruise Line - Cunard
If you are keen for some reading this Bank Holiday, settle down with this Flyertalk thread which records some of the aircraft interiors which graced the skies above the US in the early days of Jumbo jets:
TWA First Class DEN-JFK 1972 - The Good Old Days - FlyerTalk Forums