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Security sources said the Government would not pay a ransom. Ransomware is a type of malicious software that criminals use to attack computer systems.
Hackers often demand the victim to pay ransom money to access their files or remove harmful programs.
Asian countries reported no major breaches on Saturday, but officials in the region were scrambling to check and the full extent of the damage may not be known for some time.
China's official Xinhua news agency said some secondary schools and universities had been affected, without specifying how many or identifying them.
She added: 'The National Cyber Security Centre is working closely with NHS Digital to ensure that they support the organisations concerned and that they protect patient safety.' Mrs May stressed it was unlikely the hackers could access private patient data.
Experts say the cyber attack used code developed by the US National Security Agency which was leaked online last month by a mysterious group called the Shadow Brokers.
Data released under the Freedom of Information Act in December suggested 90 per cent of NHS trusts are using outdated software Windows XP, which is 15 years old and has been branded 'obsolete', leaving systems more vulnerable to attacks.
Hospitals in London, Blackpool and Colchester are thought to have been the worst hit – but last night the virus spread over the border into Scotland as the NHS admitted the extent of the chaos had not yet become clear.
It is not known how many computers were affected but if all of the NHS's 1.4 million employees were affected, the health service would have to hand over £326 million to unlock the scrambled data.
Russia is thought to be the worst affected, while Taiwan fears being the victim of a second wave.