Especially compared to Oxford, LSE.
Oxford and Cambridge have a global (and US) reputation equivalent to that of HYPS. While LSE is an outstanding university, particularly for those who want a career in finance, it is not at the same level. In some ways it could be viewed as a British equivalent to Wharton. An outstanding niche school.
As well known as Oxford, IMO.
Very - as well known as Harvard is in the UK. What is your concern?
Also, ‘well-known’ to whom? Grad schools? employers- if so, what sector/industry? Mr. Smith down in nowheresville?
^ In this case, it doesn’t really matter as Cambridge is as well-known as Oxford (so a bit more so than LSE, which has good name recognition but not the universal name recognition that Oxbridge has in the US) and is as highly-regarded by all Americans as the tippy-top American colleges (other than those, who, for some reason, put a high premium on a well-rounded liberal arts education). That would be true for Oxford as well.