Partly funded by The HELP (Helicopter Emergency Landing Pads) Appeal, a 25m by 25m helipad will enable seriously ill patients to be brought by air ambulance for treatment at the hospital's Major Trauma Centre.
A lift shaft in the middle of the wing is being built so that patients can be lowered quickly and smoothly into the hands of medical experts.
The helipad - which is due for completion in January 2014 - is being constructed by Miller Construction and will be the second hospital helipad in London and the first south of the river.
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Mayor of London Sadiq Khan is backing the scheme.
A spokesperson said the lightning strikes had left the "aircraft fuelling system unavailable."
Nearly 1,000 properties have been left without power, and road users have been warned of potentially hazardous conditions.