If you are on an a/c that has only two engines, or is not ETOPS certified (either by a/c or operating crew), then it cannot traverse the Atlantic or Pacific for that matter.
For East Coast flights returning to Britain, their routing is often as close to land as possible, however for SE US and Carribean flights, a direct routing on a 767 or 777 would have to fly up the Eastern Seaboard and take a close to land routing in case the other engine shut down or ran into trouble.
It goes to show that the event is rare, because it is extremely costly in fuel and time, but also goes to show why 4 engines are better than two, because none of this nonsense is necessary on an A340/A380/B747.
Not quite true. ETOPS operation has no direct correlation to water or distance over water. It refers to single-engine flight times between diversion airfields, regardless as to whether such fields are separated by water or land. Thus Greenland / Northern Canada are also issues - Kangerlussuaq and Søndre Strømfjord being key diversion fields available.
Furthermore, ETOPS applies to twins on routes with diversion time more than 60 minutes at one engine inoperative speed, and also applies to 4-engined aircraft more than 180 minutes diversion time.