Can British passport holders apply for a B2 USA visa from other countries, if they are living there? Or do they have to go back to Britain to apply for it?
For the most part you can apply at any US consulate. Some consulates enforce specific requirements, but these are simply to try and avoid people abusing the system in some way or other.
Given you are a current resident of the country you're in, you will have no trouble at all applying for a US visa there.
Personally I've obtained US visa both in my country of residence (Australia, whilst I was living there), my country of citizenship but not residence (Australia, whilst I was not living there), as well as a country that I was neither a resident nor citizen of (Canada).
Generally you're supposed to apply at the US consulate at your place of residence. So if you're a resident of a country you're not a citizen of (legally that is, and can prove your legal status) - you can apply at that country, regardless of your citizenship.
Otherwise they'll send you back "home" (the UK in your case), however in some places US consulates have been known to approve visas for non-residents (although from what I know it is not B2 visas, I know people who stamped their L1/H1 visas in Canada, not being Canadian citizens or residents).
To make sure - call the closest US consulate and ask them directly.
As a citizen of the UK (with the right of abode, IIRC that's how its called), you can use the WVP program and then you don't need visa at all, that provided you have a valid ESTA approval, and biometric passport, and plan on staying no longer than 90 days (without any ability to extend, under WVP its very strict).