A COBRA meeting has been held to discuss the heatwave as forecasts suggest a new record UK temperature could be set early next week.
Meteorologists have said there is an 80% chance the mercury will top the UK's record temperature of 38.7C (101.7F) set in Cambridge in 2019.
There is a 50% chance of temperatures reaching 40C (104F) somewhere in the UK on Tuesday, with the Met Office issuing its first-ever red warning for extreme heat.
The COBRA meeting, chaired by Cabinet Office minister Kit Malthouse, is the second held after the UK Health Security Agency (UKHSA) increased its heat health warning from level three to level four - a national emergency.
The Met Office has issued an amber warning for heat covering much of England and Wales from Sunday until Tuesday.
Daytime temperatures on Saturday are predicted to be around 27C (80.6F) in London, 26C (78.8F) in Cardiff, 23C (73.4F) in Belfast and 21C (69.8F) in Edinburgh. On Sunday, it could hit 30C (86F) in the capital, 27C (80.6F) in Cardiff, 24C (75.2F) in Belfast and 23C (73.4F) in Edinburgh.
Temperatures are set to increase further across the nation on Tuesday and reach the mid-thirties for much of England and Wales.
The UKHSA's level four heat health warning is reached "when a heatwave is so severe and/or prolonged that its effects extend outside the health and social care system... At this level, illness and death may occur among the fit and healthy, and not just in high-risk groups," it said.
The Met Office's red warning for Monday and Tuesday covers an area from London up to Manchester, and up to the Vale of York.
Met Office spokesman Grahame Madge said: "If people have vulnerable relatives or neighbours, now is the time to make sure they're putting suitable measures in place to be able to cope with the heat because if the forecast is as we think it will be in the red warning area, then people's lives are at risk.
"This is a very serious situation."
Train operators warn passengers to avoid anything but 'necessary' travel
A No 10 spokesman said railway speed restrictions may be needed on "some parts of the network next week to manage the hot weather and to avoid any potential damage".
Jake Kelly, of Network Rail, warned journeys will take "significantly longer and delays are likely as speed restrictions are introduced to keep passengers and railway staff safe".
Train operators have warned passengers to avoid anything but "absolutely necessary" travel on Monday and Tuesday.
Motorists have been advised to make their journeys out of the hottest periods of the day, particularly if they have older cars.
Some schools in the south of the country may close on Monday and Tuesday due to the extreme weather, and the NEU teaching union has said it will support headteachers taking this decision.
Schools choosing to close their doors have pointed to the potentially dangerous temperatures of classrooms, as well as the risk to both staff and pupils of having to work during the hottest points of day. Some schools have said they will arrange a skeleton staff to keep the buildings open for parents unable to find alternative childcare at short notice.
Some southern nurseries will also be restricting their hours on the hottest days following Public Health England's guidelines for supporting children in Early Years.
The majority of UK schools are set to break up for the summer holidays later in the week.