UK answer - I'm not sure that 'responsibility' for switching on the runway lights lies with anyone specified. ATC usually do it because they have the controls to switch it all on, off or whatever. Arguably, though, it is the airport operator that ultimately has responsibility because the international standards for lights to be displayed are in ICAO Annex 14 Aerodromes.
In the UK there are some obvious basic rules for when some lighting should be switched on - like when it's dark - but beyond that it's left to individual airports, and often individual controllers, to decide what lights to have on and what brilliancy to set. For the most part this seems to work and pilots always have the option to ask for the settings to be changed.
In the past there was more guidance in the controllers' national instruction book (MATS Part 1) about what brilliance to select for which lights in specific visibilities but that is long gone. Likewise, systems that were supposed to protect us from fools like only permitting lights to be selected for the runway that the ILS is up on seem to be going out of fashion because they limited flexibility.
Whilst HD gives you the answer for London-Heathrow, at just about every other airport in the UK the lights are switched by the controllers. And even at Heeathrow some of the lighting is switched by the controller - namely the stopbars and lead-on lighting on the main runway access points.