There are two steps involved:
Change your Siri voice settings to English (United Kingdom) - as of iOS 7.1.2 this is under Settings > General > Siri > Language and also Settings > General > International > Voice Control (they're both the same setting: change one and the other changes too).
Change your device Language setting to British English: it's under Settings > General > International > Language.
That second one was difficult to find because the default option is English - as opposed to English (United States) which always prompts me to check for an English (United Kingdom) version - and the list is not in alphabetical order: British English and Australian English are both there but way down at the bottom. And weirdly, devices bought in the UK come with most settings defaulting to British but not this one.
You'll know if your device is set to British English because the Cellular option in Settings becomes Mobile.
NB: If your device already has these settings and you still get an American accent, try changing the device language to US English, then back to UK English.