The question is rather vague, which is why it has been suggested you "google ETOP's" I suspect. This will give you an idea of the planning and operational requirements that are used specifically for twin engine extended range operations, but it will also broadly give an understanding of how all aircraft plan for various contingencies in extended range operations.
Your question states, "London to New york and more than half way through." Well that could mean all sorts of things. More than "half way through" might be near the planned destination, in which case the decision is probably a very easy one.
Does it mean half way from a distance standpoint, or halfway from a time standpoint? The answer might generate very different responses.
One thing a decompression does result in, is a descent to an altitude of around 10,000ft. The fuel use at this altitude will be significantly higher than that at the previous altitude. This is one of the specific planning considerations for an extended range dispatch. You must obviously always have fuel to safely divert to an alternate airport at any point in the flight.
The weather at 10,000ft can be a more significant factor than it is at cruising altitudes, and this needs to be taken into account.
Passengers may as a result of the incident need medical attention that in itself necessitates a diversion.
The aircraft itself may have structural or mechanical issues that are causal to the decompression, that necessitates a diversion.
Add these factors up: Fuel, weather, safety and technical, and it is clear that you are going to divert to a planned adequate and suitable airport where your departure or destination airport is not geographically already the best option.
Each extended range flight has it's own specific route considerations that need to be taken into account at both the planning and operational stages.
In the example you give, London to New York, you should remember that this isn't two cities seperated by a vast body of ocean (as people sometime perceive.) Notwithstanding Westerly winds, Jetstreams, etc. The Great circle distance form London to New york is 3451 miles. However only 1980 miles of that is across the open ocean. The great circle route (which on any given day may not be the shortest flight time route,) would take you almost directly overhead Shannon in Eire and Gander Newfoundland in Canada. As such the first 370 miles is largely over a land mass with a lot of potential diversion airports. The last 1100 miles is also over the land mass of Eastern Canada and the Northwestern United States, with similar potential diversionary options.
Distance wise, the half way point on this trip would only be around 600 miles East of Gander or other Newfloundland airports. Time wise (with prevailing Westerly winds) half way would be even closer to these airports.
There are many fixed considerations, that will apply to the planning and operational contingencies for every commercial flight. There are also many dynamic considerations that change on an hourly basis, that also need careful consideration. Fuel loading, weather, airport availibility, aircraft weight and performance, mechanical, medical. The list goes on.
This is a subject that could fill pages and still fail to cover every option, consideration or contingency, but I hope this answer has given you some insight and food for thought.