Nope, you can't link a non-UK based account with the UK Kindle Store. You can't even *buy* the Kindle itself on Amazon UK, only accessories
. (In fact, I'd advise buying the covers on Amazon UK anyway, there is (or was, it may be 'fixed' now) a crazy import fee to France on the covers when buying from Amazon US).
If you happen to have a UK address/credit card/id/account, there should be no problem, though.
And I'm not sure if we have a subset of the UK content in the french Kindle Store, last time I checked was before the UK store
EDIT: For the few things I checked, yep, it seems that we do (or a subset of it, at least). At some truly horrendous prices in some cases (Hi, Hachette! Bye, Hachette! >_<"), but they're available, yay!