The National Crime Agency is warning computer users they have two weeks to protect against a "powerful computer attack".
It comes as US officials held a press conference accusing a Russian hacker of masterminding the scam and raking in £60m.
Two pieces of malware software known as GOZeuS and CryptoLocker are at the centre of the alert.
People are being warned to make sure their security software and operating system are both up to date, and to run scans to check for any problems.
Important files should also be backed up, said the UK's National Crime Agency (NCA).
The malware typically infects a computer via attachments or links in emails.
If a user clicks on one of them, the malware silently monitors activity and tries to capture any private information, such as banking details.
"(The links or attachments) may look like they have been sent by genuine contacts and may purport to carry invoices, voicemail messages, or any file made to look innocuous," the NCA warned.
"These emails are generated by other victims' computers, who do not realise they are infected, and are used to send mass emails creating more victims."
The agency said GOZeuS (also known as P2PZeuS) was responsible for hundreds of millions of pounds of fraud globally.
Sky's Technology Correspondent Tom Cheshire said: "We should all be concerned. It goes specifically after financial information, and if that is over a certain threshold it starts stealing it very silently without you necessarily noticing."
The second threat comes from the Cryptolocker malware, activated if the first attack is not profitable enough.
It locks a user out of their files, for example photos or music, and threatens to delete them unless a "ransom" of several hundred pounds is paid.
More than 15,500 computers in the UK are infected and "many more" are at risk, according to the NCA.
Stewart Garrick, a senior investigator with the NCA, told Sky News the threat was mainly against individuals or businesses running Windows-based computers.
"It's utterly indiscriminate - it doesn't care what machine it falls on.
"It's there purely for the criminals to harvest your money direct from your bank account, or by demanding a ransom to get your files back."
FBI action in the US had weakened the network of affected computers, said the NCA, "meaning that action taken now to strengthen online safety can be particularly effective".
Thirty-year-old Russian Evgeniy Bogachev is the alleged leader of the gang behind the attacks, FBI executive assistant director Robert Anderson told a news conference in Washington DC.
"They (the FBI) have disrupted the network and taken control of it," said Sky's Tom Cheshire.
"So when the hackers try to speak to the computer that's affected, that line of communication has been cut off.
"You now have a chance to clean up. The first thing you should do is update your operating system - especially if you're on Windows, then look to scan your computer for viruses and it should be able to find it."
For more information visit www.getsafeonline.org/nca.