A feared second spike in the attack failed to materialize on Monday but the healthcare system was still disrupted.
The new infections were largely in Asia, which had been closed for business when the malware first struck.
"We have not seen a second wave of attacks and the level of criminal activity is at the lower end of the range that we had anticipated and so I think that is encouraging", he told Sky News.
Spanish firm Telefonica, French automaker Renault, the US -based delivery service FedEx and the German railways Deutsche Bahn were among those affected.
Companies and government agencies in the region are on high-alert and say it is only a matter of time before the current ransomware wave of attack infects computer systems in the region. Europol said Monday that "very few" people have paid the ransom, and security company Bitdefender said the attack has raised just over $51,000.
Cyber security experts in the National Health Service (NHS) worked alongside the National Cyber Security Centre (NCSC), part of the GCHQ spy agency, to patch computer systems after the attack caused widespread problems on Friday, Mr Wallace said.
In a statement, the Home Secretary, Amber Rudd, denied that under-investment in cybersecurity had caused the problem.
How did the attack spread worldwide?
Microsoft released a security patch for the vulnerabilities in March.
"The latest count is over 200,000 victims in at least 150 countries, and those victims, many of those will be businesses, including large corporations", he said.
However he has said he is now concerned that "terrible things" may be done in retaliation for his efforts. "It is so visible and so global". Staff are working hard to ensure that the small number of organisations still affected return to normal shortly.
In China, "hundreds of thousands" of computers at almost 30,000 institutions and organisations were infected by late Saturday, according to Qihoo 360, one of China's largest providers of antivirus software.
Chief executive Paul Hawkins said: "As a precautionary measure over the weekend a comprehensive programme of diagnostics and preventative work was carried out to protect our systems".
"Microsoft's leadership stated this directly, they said the source of the virus was the special services of the United States", Putin said on the sidelines of a summit in Beijing.
If you're not on a work network that already has security, consider installing some form of security program on your computer. The malware spread through phishing attacks, malicious emails and infected attachments, encrypting every file it can on a computer and then posting a landing page demanding a $300 ransom payment in Bitcoin in order for the files to be unlocked.
Another obvious defence is to install anti-virus software that scans files before they're downloaded.
Ryan Kalember, senior vice president at Proofpoint Inc., which helped stop its spread, said the version without a kill switch could spread.
Once the computer's data have been encrypted, or "locked down", a message would appear saying that the users' files are no longer accessible.
"We don't expect this to be a sophisticated group", said Wellsmore.
AP however reported that Microsoft's top lawyer is laying some of the blame for the massive cyber attack at the feet of the U.S government.
The VNCERT, under the Ministry of Information and Communications, has issued warnings as well as offered protection measures to all users to guard against the ransomware and its variations, which target Microsoft Windows - an operating system that is widely used in Việt Nam, especially the outdated Windows XP. Cybersecurity experts say the unknown hackers who launched the attacks used a vulnerability that was exposed in NSA documents leaked online.
Mrs May said the Government was putting £2 billion into cyber security. Bossert said that while USA officials had not ruled out the possibility that it was a "state action", he said it appeared to be criminal in nature, given the ransom requests.
But Europol warns that paying up doesn't guarantee that you'll get everything back. "You can change the locks but what has happened cannot be undone".