The East Coast Main Line will close on Tuesday as temperatures are forecast to hit up to 41C (106F).
A number of rail services have been cancelled or restricted as the country braces for extreme heat.
East Midlands Railway is warning of cancellations or reductions to its services from Sheffield and Lincoln. Hull Trains said it would be only running a single service early in the morning in each direction between the city and London on Tuesday. Speaking to BBC Radio 4's Today Programme, Mr Kelly said: "Our advice very strongly to customers in England and Wales today and tomorrow is to only travel if absolutely essential, and to expect a very reduced train service and delays.
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