Prefer Non-Oxbridge Applicants

Are there any universities that prefer applicants who have not applied to Oxbridge (which they can tell of course by the date of application)?

No they don't. THey can assume but they do not know. I think I sent my appication off before the Oxbridge deadline (I applied to Durham, Glasgow, Newcastle and Leeds originally). It was eight years ago so memory might play tricks. It was certainly very early and well before everyone else. Late(ish) September.

Officially an applicant's other choices are secret until a firm or insurance is chosen.

Universities haven't had access to the identity of your other choices for about 20 years. Before that there were many rumours of petty jealousy between institutions e.g. Kent wouldn't offer you a place if you put them below Leicester. Durham was fairly notorious. The fact they changed the rules suggests there was something in it.

Someone once told me that if you applied to King's then UCL wouldnt give you an offer and vice versa. .

(which they can tell of course by the date of application)

What difference does it make if most people who apply for Oxbridge don't get in?

People are apprehensive of applying before the Oxbridge deadline to non-Oxbridge universities because they fear these universities will infer from the date of application that the applicant is an Oxbridge-applicant. Thus they believe the university will treat their application unfavourably on the basis that the applicant is likely to go to Oxbridge and turn down the university's offer.

However, seeing as most Oxbridge applicants are turned down, other universities will be aware of the fact that only a very small proportion of those applying before the Oxbridge deadline will turn their offer down for a place at Oxbridge (due to the additional numbers of non-Oxbridge applicants just filing a timely application which happens to fall within the Oxbridge deadline). And as such, the benefits to a university of separating likely Oxbridge applicants from other applicants becomes marginal and ultimately pointless.

If applying early meant almost definitely gaining admission at Oxbridge, there would be good reason to hide that fact from other universities, for obvious reasons.

But what about LSE for Economics? Maybe they would think that if they were worthy of an LSE offer, they would most likely get an Oxbridge offer too?

Applying and being accepted are entirely different things.

People are apprehensive of applying before the Oxbridge deadline to non-Oxbridge universities because they fear these universities will infer from the date of application that the applicant is an Oxbridge-applicant. Thus they believe the university will treat their application unfavourably on the basis that the applicant is likely to go to Oxbridge and turn down the university's offer.

However, seeing as most Oxbridge applicants are turned down, other universities will be aware of the fact that only a very small proportion of those applying before the Oxbridge deadline will turn their offer down for a place at Oxbridge (due to the additional numbers of non-Oxbridge applicants just filing a timely application which happens to fall within the Oxbridge deadline). And as such, the benefits to a university of separating likely Oxbridge applicants from other applicants becomes marginal and ultimately pointless.

If applying early meant almost definitely gaining admission at Oxbridge, there would be good reason to hide that fact from other universities, for obvious reasons.

I think it usually wouldn't make a difference. There might be a case for saying that places like Durham, LSE or UCL wouldn't offer too many places to those who applied early, 'just in case' they then didn't have enough places filled.