I'm not sure London to New York would cause too many problems. London is about ten degrees further north than New York, which means that, in theory, the sun will be slightly higher in the sky in New York at the same time of day, at the same time of year. In practice, I wouldn't expect that to be something that will be noticeable to anyone.
London gets neither as hot nor as cold as New York, thanks to the Gulf Stream. Most winters in London will not look much like the same time of year as New York. It won't be as bitterly cold, and snow is extremely rare in London. Summers may look similar, although London will not generally get quite as humid. The London weather is not actually as bad as it is reputed to be - actually, New York gets more rain, more bitter cold, and more sweltering humidity - but London can be extremely unpredictable, especially in spring, to the point of being very troublesome for filmmaking.
It isn't clear in your post whether you're trying to make London look like New York, or New York look like London. My personal experience of New York is limited to a few weeks here and there, but your principal concern is that the two cities really look nothing like one another. You may find a few tiny vignettes that can pass either way, but for a convincing result it will need dressing with a significant amount of stuff, depending on the shot - vehicles, street furniture, etc. I don't know, but I'd hazard a guess that it's probably somewhat easier to rustle up some American cars and fire hydrants in London than it would be to do the opposite in New York. It really depends what you need. If you want the exterior of a glass and steel office building, well, they're pretty international. If you want a full blown street scene, the cost of simulating it could rapidly escalate beyond the cost of moving a small production to the alternate city.
In terms of cinematography, my feeling is that most people tend to depict both as bluish, whereas more southerly locales - California, the Mediterranean coast of Europe - are depicted in warmer tones. Separating the two northerly cities based on light might be hard, and if you want accuracy, it's been my observation that the typical steaming vents of New York are rapidly being remediated out of existence.
Put a yellow cab in shot, perhaps? The entire "driving to work" sequence in Spy Game was shot in London doubling for Washington, DC, although there were a couple of quite noticeable glitches in it. And they made a historic part of Oxford look like a Chinese prison, but then Chinese prisons aren't probably that well known to the core audience.