Having vectored good numbers of commercial aircraft with engine failure (a fair percentage with 50% loss of power) over central London I often wondered about this matter.. However, during my time at Heathrow there were no specific directions to controllers to direct such aircraft on to other than the usual flight paths - may there are such procedures now? For landing on westerlies at Heathrow it would be exceptionally difficult to avoid "built up" areas, which are in some profusion to the east of the airport. And what does one do about the Airbus I once had which did two missed approaches in LVPs and lost an engine on the second. He diverted straight to Manchester on one engine, thereby overflying "built-up" areas much of the way.
ATC radars do not show terrain in any form so it would not be possible to vector aircraft clear of built-up areas with any degree of accuracy.
There was (and probably still is) a procedure for the Airport Authority to request, via ATC, commanders of aircraft with technical problems(s) to consider diverting elsewhere. But, of course, that decision rests with the commander.