Whilst or while, etc - Oxford Spelling

It's more important to be consistent (even if you're consistently using while for one context and whilst for another), but in short the answer is

While

Oxbridge English is of course definitionally British English and, if anything, the prestige dialect. The house style of Oxford University Press (as well as Nature) is likewise definitionally a form of British English.

Your specific examples are understandable but badly taken. The spelling civilization is more common than civilisation in British publications (in addition to being more etymologically & phonetically accurate, not that any form of English ever really cares about that); the spelling while is vastly more common than whilst in British English (and always has been); the spelling among is just as vastly more common than amongst in British English (and likewise always has been). There's a general sense that civilisation, whilst, amongst, stone, tonne, & all are non-American but, whilst that may be helpful to marketers trying to make their products sound especially Britishy, it doesn't necessarily reflect the actual speech of actual Britons.

In any case, even though that should already be enough to reconcile you with Oxford's "Americanized" British English (which was always just its own), the 2014 style guide already linked above and the present (2016) version available from the University's Public Affairs site noticeably use while and among repeatedly while eschewing either whilst or amongst. As Mr Ashworth already noted, that's in keeping with their own guidance to write as tersely and clearly as possible. Similarly, using Google to search crawled sites available at the domain www.oup.com, there are occasional instances of whilst but many, many more of while.

Which would be correct to use according to the Oxford spelling?

The OED gives

whilst, adverb and conjunction (and preposition) = while conjunction (and preposition)

The adverbial use is marked as "obsolete or dialect." Hence the words are the same.

The "rule" is thus "Use whichever you want, but be consistent."