This question was inspired by the comment/chat discussion under this answer. This would be very helpful to know.
The problem with using the 'British English' corpus selection on ngrams is that the corpus includes American books that were published in the UK. It is not a selection of British texts, so it does not accurately represent English as used in the UK.
This ngram for fantasize vs. fantasise is misleading. UK spelling convention is fantasise despite the ngram and whatever was written in the original question thread.
The BNC (British National Corpus) is a 100 million word collection of samples of written and spoken language from a wide range of sources, designed to represent a wide cross-section of British English.
Excluding spoken English, a search of the BNC revealed 34 counts of fantasise, but only 16 for fantasize, of which 8 are from the American magazine Esquire. Regardless of dates, this clearly disagrees with the ngram.
The 'British English' google corpus can still be useful in comparing relative usage frequencies between British and American English over time, but care should be taken when interpreting the results, and I certainly don't think it should be relied on to settle matters of British English usage.