Hospitals are notoriously slow in applying security fixes, in part because of how disruptive it is to take patient-facing equipment and databases offline.
Home Secretary Amber Rudd said 48 NHS trusts were affected and all but six were now back to normal.
Rudd said the attack was not specifically targeted at Britain's health service.
A Nissan UK spokesman confirmed it was first affected by the attack on Friday night but there had been "no major impact".
Researchers with security software maker Avast said they had observed 57,000 infections in 99 countries, with Russia, Ukraine and Taiwan the top targets. Both said Russian Federation was hit hardest.
Cybersecurity experts identified the malware as a variant of the ransomware known as Wcry or WannaCry.
Security experts warn there is no guarantee that access will be granted after payment.
Images appeared on victims' screens demanding payment of $300 (275 euros) in Bitcoin, saying: "Ooops, your files have been encrypted!"
Authorities in both countries said the attack was conducted using "ransomware" - malicious software that infects machines, locks them up by encrypting data and demands a ransom to restore access. It was always going to happen. "There's no barrier to do it tomorrow to 100 million computers".
Europol, the European Union's police agency, said the onslaught was at "an unprecedented level and will require a complex worldwide investigation to identify the culprits".
The malicious software behind the onslaught appeared to exploit a vulnerability in Microsoft Windows that was supposedly identified by the National Security Agency for its own intelligence-gathering purposes and was later leaked to the internet. It was reportedly distributed by the Shadow Brokers, which claimed to have hacked an NSA-linked team of hackers last August.
The researcher, tweeting as @MalwareTechBlog, said that the discovery was accidental, but that registering a domain name used by the malware stops it from spreading.
This means that it gets into your computer and looks for other computers to try and spread itself as far and wide as possible.
The Government has invested to protect against a cyber attack on the NHS, ex-NHS Digital chairman Kingsley Manning said, but he added that it can be "difficult" to ensure trusts spend on cyber security. Other impacts in the USA were not readily apparent on Saturday. Computers already affected will not be helped by the solution. Short of paying, options for these individuals and companies are usually limited to recovering data files from a backup, if available, or living without them.
Security experts said the attack appeared to be caused by a self-replicating piece of software that enters companies when employees click on email attachments, then spreads quickly internally from computer to computer when employees share documents.
British media had reported previous year that most public health organizations were using an outdated version of Microsoft Windows that was not equipped with security updates.
"This is a virus that attacked Windows platforms". Microsoft says now it will make the fixes free for everyone.
She said it was "disappointing" some hospitals had been using the outdated Windows XP operating system, despite being instructed not to by the Health Secretary Jeremy Hunt.
At two London hospitals, a British medical student found widespread computer issues, he told CNN.
"As of now the virus has been localized", ministry spokeswoman Irina Volk told TASS. Russia's health ministry said its attacks were "effectively repelled". Officials played down the incident, saying the attacks had been contained.
Russian cellular phone operators Megafon and MTS were hit.
Britain's National Cyber Security Centre and its National Crime Agency were looking into the United Kingdom incidents, which disrupted care at National Health Service facilities, forcing ambulances to divert and hospitals to postpone operations.
The country's central bank said the banking system was hit, and the railway system also reported attempted breaches. Radio Slovenia said Saturday the Revoz factory in the southeastern town of Novo Mesto stopped working Friday evening to stop the malware from spreading.
Thirteen Scottish health boards have been hit by a global cyber attack that affected NHS computer systems.