You should get into all bar Oxford and LSE. LSE is mostly about grades but significantly more competitive than your other choices, but you have a good chance (would say >50% likely). You’ll most likely get an interview at Oxford providing your test is good, and then most of whether you get it comes down to that interview.
Any reason you’re not applying to UCL? Job prospects wise it’s a big step up from Durham and KCL and only a marginal step below LSE, but it will be a lot less competitive to get into (would say you’re pretty much guaranteed providing personal statement and references are good).