Is Flight Following in the USA directly equivalent to Basic Service in the UK?
Based on the UK Flight Information Services document that Dan helpfully provided in chat, it looks like they're different:

Basic Service


Available to IFR flights in class G
Available to VFR flights in class E and G
Provides information on "airspace activity that may affect your flight"
Pilot is free to manoeuvre as needed, unless otherwise "agreed" with ATC
Provided workload permitting


Flight Following


Not available to IFR flights
Available to VFR flights in any airspace where ATC has radar coverage
Provides information on airspace, traffic and weather
Pilot is free to maneuver as needed, unless otherwise instructed by ATC
Provided workload permitting


So the major differences seem to be:


Basic Service is available to IFR flights; Flight Following is not (it wouldn't make sense anyway)
Basic Service provides airspace information only, not traffic information; Flight Following provides both


The whole traffic vs airspace information thing is very confusing in the CAA doc. Read literally, there's no VFR service that offers both airspace and traffic advisories, but that makes no sense at all. I'm probably missing something, but reading that document makes me glad the US has kept it simple...