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It looks like you cannot apply for a Schengen visa in London.
To apply for a visa from the French Authorities in London:
- You must be resident in the UK
Your UK residency must have at least 3 months validity after the intended date of departure from the Schengen area. As per the decree 2008-1176 of the 13th of November 2008, the French Consulate, London, is not competent in processing visa applications of people not resident in the UK, i.e. holders of βC-Visitβ visas or of a British visa valid for 6 months or less. People who are not resident in the UK must apply for their Schengen visas at the French Consulate in their country of residence.