While I'm sure that must be some lovely local communities somewhere within London, it's long been regarded as the unfriendliest city in the country in general.
A move elsewhere might, indeed, be a sound idea but you need to ensure that you choose a city (or town/village) that's known for its warmth. That won't guarantee that you'll bond with your neighbours but it will certainly improve your chances of doing so. For example, I wouldn't recommend moving here to Suffolk, where you usually need to live at one address for at least a decade before any of your neighbours are bold enough to wish you 'good morning'. (I've lived here for over thirty years. I'm on 'nodding terms' with just one neighbour, who's been here for nearly as long, but I don't know anyone else in my street at all. I rather like it that way but I don't think that Suffolk would be for you. Norfolk is probably even worse!).
The friendliest city in the south of England is almost undoubtedly Brighton, where the large gay community seems to contribute to the city's warmth. Further north, Sheffield (where I lived for 20 years) is an amazingly friendly city, as is Liverpool.