Southbound trains are running only as far as Darlington, while northbound trains are terminating at Doncaster.
Passengers are being shuttled the 90 miles between the stations by coach.
The 08:30 Edinburgh to Kings Cross service terminated at Newcastle due to the signalling issues as did the 09:00 service.
London North Eastern Railway (LNER) said it hoped to restore a “much rdeuced and slower” through service later today.
But its advice to passengers remained: “Do not travel today”.
Tickets can be used instead on Virgin Trains services on the Glasgow-London west coast main line.
A LNER spokesman said: “Severe weather in Yorkshire has affected all signalling in York and Leeds.”
A Network Rail spokesman said: “A signalling cabinet was struck by lightning outside York station which is causing significant disruption.
“A lot of trains are at a standstill including everything around York.”
Paula Dunstable tweeted: “Seems a lightning strike has taken out the whole [East] Coast Mainline ... stuck at Peterborough... on an increasingly hot train the LNER 09:30 Kings X /Edinburgh going nowhere ... with 400 other passengers.
“@LNER staff doing their best and being brilliant... ice and water.”
The disruption comes as the UK experiences soaring temperatures on what has been dubbed ‘Furnace Friday’.
Forecasters have predicted a high of 27C in Edinburgh, with temperatures expected to be in the mid-30s for much of east and south-east England, reaching 37C in London.
A Met Office yellow “be aware” warning for thunderstorms has been issued for Central, Tayside and Fife, which will come into force from 02:00 to 14:00 on Saturday.
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