How much power do planes use when taking off at London City?
Let's say there is an infinite runway, and a twin-jet capable of flying the whole route on one engine.
The jet starts its takeoff roll at maximum thrust, and at 40 knots an engine quits. Does the crew continue or not?

If they continue, they'll veer off the runway. Because the asymmetric thrust is huge and the airspeed is low, they won't be able to keep the plane on the centerline with the rudder.
Enter Vmcg

Vmcg is defined as the minimum speed, whilst on the ground, that directional control can be maintained, using only aerodynamic controls, with one engine inoperative (critical engine on two engine airplanes) and takeoff power applied on the other engine(s).

Why is it important?
The lower the thrust on short runways, the more you can carry...
It may be counter-intuitive, but here's why: there are three types of thrust reduction on takeoff:

Derate
Assumed temperature (aka flex)
Derate + flex

(Availability of the modes depends on the options available for the type and the options chosen by the airline for their fleet.)
Derate vs. flex and effect of derate on takeoff weight


The minimum control speeds associated with the FLEX take-off are related to VMCG/VMCA at TOGA thrust. Therefore, should the aircraft suffer an engine failure at V1, there is no limitation on selecting TOGA thrust on the remaining engine(s).
DERATED THRUST TAKE-OFF
A reduction in take-off thrust leads to lower VMCG/VMCA and hence, a lower V1. When taking off from short or contaminated runways where ASDA [Accelerate-Stop Distance Available] is the limiting factor, a reduction in the minimum control speeds may generate a take-off performance benefit and a higher MTOW.
As opposed to a FLEX take-off, the selection of TOGA following an engine failure during a derated thrust take-off is prohibited as long as speed remains below the first flap/slat retraction speed for the take-off configuration. The selection of TOGA below this speed in these configurations may result in loss of control due to VMCA considerations.
— A330/A340 Flight Crew Training Manual

Taking off from LCY
A BA A318 taking off from LCY headed to New York with lots of fuel might as well be using reduced (derate) thrust because of this shorter runway.
In short, it's very hard to compare even if the only variable is the runway length.
Flex example
The CFM56-5B3 jet engine (powers the A321) can reduce thrust by 35% when a Δ35°C assumed temperature is used. So 120 kN of thrust becomes 78 kN (per engine).

Further reading:

Airbus: Flex/Derate, Engine Bump and Derated climb
CFM: A320 Flight Ops Presentation