An American Airlines Boeing 767-300, registration N397AN performing flight AA-199 from Milan Malpensa (Italy) to New York JFK,NY (USA), was enroute at FL340 about 190nm westnorthwest of Shannon (Ireland) and about to enter the Oceanic Crossing when the crew reported cabin problems and received delay vectors to avoid entering the North Atlantic Track. After working the related checklists the crew advised they had problems with the cabin pressure control that they could not do the Atlantic Crossing with, advised they were planning to divert to London Heathrow,EN (UK) and requested to descend to FL240 or so. The aircraft was cleared down to FL230, turned towards Heathrow and landed safely in London Heathrow about 100 minutes later. The aircraft remained on the ground for about 140 minutes, then departed Heathrow and continued to New York where the aircraft landed with a delay of 5 hours.