What procedures are in place at London City Airport for emergencies after V1?
London City certainly has its challenges, with a steep approach and high rise buildings in close vicinity to the short runway, at the same time you have all sorts of places to divert to: Southend to the east, Stanstead and Luton to the North, Heathrow to the west, Farnborough, Gatwick, the list goes on. All of these places have longer runways with less constrained areas and fields all around. North Weald and Biggin Hill could also be used in an emergency. By the time you have returned to LYC you could have reached one of those. 

It depends on the severity of the situation; if you have a mechanical problem but can land safely then going back to LCY is convenient for your passengers, if you're on fire or have control problems you're better off an another airport. 

It is certainly possible to return to the airport in an emergency, and it has been done at least once. In 2014 an Avro RJ85 returned to the airport after a part of a nacelle came off. The headline is misleading, it says part of the wing fell off but it was really a nacelle. This is sort of "emergency-lite" as the flight could not continue, but the condition of the airplane was good enough that a normal approach could be made and the passengers moved to another flight. 

The same year an Avro RJ100 diverted to Stanstead after an engine fire. 

So the wisdom of returning to London City in an emergency is entirely dependent on context like the condition of the aircraft, weather, etc.