Carole Souter, chief executive of the Heritage Lottery Fund, said that the cash would also help promote tourism:
"We know that heritage is a huge draw for visitors from home and abroad.
"More than a quarter of all UK holiday activities undertaken by UK residents now involve heritage.
"These projects all offer the public the chance to explore and enjoy our rich and complex history."
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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.