What Are Some Private/Ivy League Colleges in the U.K.?

Firstly "Ivy League" is an American term which has no equivalent in the UK. Secondly, the term "school" in Britain is exclusively used for primary and secondary level educational bodies i.e. no later than the equivalent of High School; tertiary level bodies are all called either universities or colleges - some universities may be comprised of "schools" for individual subjects, but most people wouldn't recognise the term as referring to tertiary level education in general. Except for the University of Buckingham, all universities in the UK are primarily government funded, though tuition fees are usually charged (except for Scottish students attending Scottish universities) with higher fees charged for foreigners (who are therefore very attractive to the universities, as they bring in more money). What is the best university will depend on what subject you're studying, but in general, the older the university the more prestigious it is - so you start with the medieval universities Oxford, Cambridge, Glasgow, Edinburgh, St Andrews, and Aberdeen (in the sixteenth century the Scottish city of Aberdeen alone had as many universities as the whole of England!), then the nineteenth-century and early twentieth-century universities like London, Durham, Birmingham, Liverpool, Manchester, Leeds, and Sheffield. Then there are the "New Universities", a few founded in the 1920s like Reading or Bristol, and then a wave built in the 1960s - York, East Anglia, Essex, Sussex, Lancaster, Kent. From 1992 onwards there has been a huge wave of former polytechnics and other institutes which have been given university status. There are a few anomalies, like the universities of Aberystwyth, Bangor, Cardiff, Swansea and Lampeter which have only existed as separate institutions for the last few years, but had been constituent colleges of the federal University of Wales since the nineteenth century.

There is only one answer: Cambridge.