There is some misinformation on this thread about the effect of state vs independent education on Oxbridge applications.
Selection of university applicants is mostly done on quantifiable data and the whole process has to be transparent.
Effect of applying from a state school:
When universities look at the GCSE grades of an applicant, many will compare these against the average GCSE grades in the school where the GCSEs were taken. In very selective schools with very good GCSE results, the applicant is going to have to get much higher grades to stand out against the average. The applicant would have this problem whether they attended a super selective state school or an academically selective independent school. Applicants from a comprehensive school with low attainment 8 perform comparatively better on this measure (in an attempt to even the playing field slightly).
The comparison data is taken from the school where they took their GCSEs, not the sixth form they currently attend, so moving to a state school after GCSEs has no effect.
Outreach days and summer schools:
For some summer schools (Sutton Trust) the applicant has to tick several contextual boxes (school with lower than average GCSE results, living in area of low progression to HE). For others (UNIQ, some outreach events) anyone from a state school can apply.
Contextual offers:
The markers used for this are usually based on rates of progression to higher education within your postcode (e.g. POLAR). Oxford and Cambridge may invite extra candidates for interview based on contextual markers, but the offer given will usually be the same as for any other applicant.
TL;DR
You are more likely to get an interview if you went to a comprehensive with low GCSE results or if you live in an area where hardly anyone goes to university. Other than this, neither Oxford not Cambridge will take into account what school you went to.
There are several factors to take into account when choosing a secondary school, but the difference between these two schools is not relevant for Oxbridge applications.