Several drownings were reported in Britain and France as people tried to cool down.
London recorded a rare "tropical night", defined as one in which the temperature does not fall below 20C, and Britain's Met Office weather service said the temperature in southern England could reach 35C on Tuesday.
Monday was the UK's hottest May day on record, with the temperature hitting 34.8C at Kew Gardens in London, smashing the previous record of 32.8C set in 1922 and 1944.
Records also fell in France, where temperatures reached 36C on Monday and widely remained above 20C at night.
The national weather service, Météo-France, said a "heat dome", with heat held in place by a high-pressure weather front, was producing temperatures more than 10C above what used to be usual for this time of year.
Unpredictable and extreme weather are becoming more frequent as earth's warming builds.
Experts say unprecedented and deadly weather extremes that sometimes strike at abnormal times and in unusual places are putting more people in danger.
After a UK long weekend that sent people flocking to beaches, pools and shady parks, London commuters sweltered on Tuesday in subway carriages without air conditioning.
Trains to and from the busy Waterloo station were disrupted by a report of smoke on the tracks.
In Scotland, firefighters worked through the night to douse a grass fire that sent smoke billowing from Arthur's Seat, the rocky hill that looms over Edinburgh.
The UK Health Security Agency issued an amber health alert for large parts of the country through Thursday, warning of a potential health risk, particularly among older people, at the hottest times of the day.
The UK is used to moderate temperatures, and many homes, schools and businesses do not have air conditioning.
A 13-year-old boy died in the water of a reservoir in Halifax, northern England, on Monday, police said.
French government spokesperson Maud Bregeon said there were reports of at least seven deaths potentially related to high temperatures, including five drownings and two deaths in sports competitions.
The early heatwave has struck before the annual summer window when lifeguards watch over bathers at popular beaches, increasing risks.
On France's Atlantic seaboard, where magnificent beaches have powerful riptides, officials reported a rash of emergencies in the surf with two drowning deaths on Sunday at popular resorts in the Gironde region in the southwest.
The unseasonable heat extended to Spain, where weather service spokesperson Rubén del Campo said "we find ourselves with temperatures we normally see in the middle of the summer now in the month of May".
14°C