London Universities do find it difficult to attract British students from outside London and the SE, especially those who are not from ethnic minorities. (Those from private schools being the exception.) Issues around cost and concern about "University experience". The richer Universities, especially Imperial, do what they can do what they can in terms of bursaries, but it is what it is. If British students decide that "the Chinese" won't mix, so be it.
London Universities have a good track record for outreach from less advantaged London schools/London populations. Often the target pupils will prefer Imperial over Cambridge. Partly for cost, but also because they offer more of a known environment.
The Chinese diaspora is enormous and "the Chinese" can be quite diverse. Amongst DS' friends one came from mid-Wales where his family owned a restaurant and who was only too keen to be somewhere where he was not different. Another came from an outer London suburb. A third, with an affluent Hong Kong background, had attended a major public school. A fourth had a father who worked for an American multinational so had grown up away from China. And so on. Two of his close friends were second generation Londoners (one Polish, one from SE Asia) whose parents were manual workers. My own experience of LSE, many many years ago, and even then the only Brit on my course, is that the diversity can be enriching, but you need to embrace it. It is not for everyone. Oddly a regular complaint from London students who study elsewhere is that places like Bristol can feel quite monocultural.