There is a rumour/myth/experience that LSE in particular and some other universities in UK tend to reject applicants who apply to Oxford or Cambridge - i.e. submit their application before 15th October.
We are not in a position to tell you whether this is true or not - certainly we know of applicants who were rejected by all university choices except Oxford or Cambridge, whilst others got offers from universities in London and across the country as well as Oxford or Cambridge.
You can avoid this dilemma/insecurity by submitting your UCAS application for Oxford or Cambridge by October 15th and listing only one university (Oxford/Cambridge). UCAS allows you to add other universities later, so you can add LSE and other London and UK universities later in the year or in early January, in time for the January 15th deadline.
LSE tends to make its offers later in the year - in early December or after the Christmas break - so you can submit your application there in November to simultaneously take advantage of not making your application to Oxford or Cambridge as obvious, yet still being among the first applications they view and send offers to.
An alternative explanation for admission at Oxford or Cambridge despite a rejection from LSE could, of-course, also be that the universities differ in what they look for in candidates, and that Oxford and Cambridge take more and different information into account, including interviews and (usually) tests.
Katharina Kunze, Tutorin und Gründerin von Oxbridge Bewerbung, hat zwischen 2005 und 2008 an der Oxford University "Politics, Philosophy and Economics" (BA Hons) studiert.
Nach ihrem Master hat sie als Unternehmens-
beraterin bei Oliver Wyman, bei The Economist und bei dem deutschen Startup mymuesli gearbeitet.
Seit 2011 haben sie und ihre Mentoren bei Oxbridge Bewerbung Schüler in England, Deutschland und im Ausland bei der Bewerbung für Oxford und Cambridge University betreut.