O2 said it did not know when full service would be restored to millions of customers who cannot access data services on their smartphones.
The mobile network said it was working "flat out" with Ericsson to fix the technical problems.
The Swedish company, which makes mobile network equipment, confirmed that an issue with its software was to blame.
Services such as bus timetable information have also been affected, while many businesses faced disruption.
O2 has 25 million customers and also provides services for the Sky, Tesco, Giffgaff and Lycamobile networks, which have another seven million users.
"The network issue we are currently experiencing remains a top priority for us and we are working as quickly as possible to fix the issue," O2 said.
Its chief executive, Mark Evans, tweeted to "reassure our customers that we are doing everything we can to fix the issue with our network".
Skip Twitter post by @MarkEvansO2
I want to reassure our customers that we are doing everything we can to fix the issue with our network and say how sorry I am to everyone affected. My teams are working really hard with Ericsson to find a swift resolution. Stay updated: https://t.co/TGw5OUurma
— Mark Evans (@MarkEvansO2) December 6, 2018
End of Twitter post by @MarkEvansO2
Ericsson said there had been network disruption for customers in multiple countries.
President Börje Ekholm, said: "The faulty software that has caused these issues is being decommissioned and we apologise not only to our customers but also to their customers. We work hard to ensure that our customers can limit the impact and restore their services as soon as possible."
Network services on most of the affected networks had been restored on Thursday, he added. "We are working closely with the remaining customers that are still experiencing issues."
Japan's Y!Mobile network, owned by Softbank, was one also affected by the outages.
The difficulties in the UK were first reported at about 05:30 on Thursday.
Network down: how people are coping without a signal
Spain's Telefonica owns O2, which has the UK's second-largest mobile network after EE, which is now part of BT.
The company has said voice calls are not affected by the problem, but some O2 customers say they cannot make calls or send texts either.
The outage is having knock-on effects for other services that use the O2 network, including Transport for London's electronic timetable service at bus stops, which has stopped working.
William Webb, a tech consultant and former Ofcom director, said it was unusual to take this long to resolve the problem: "It's probably some kind of a software update that one of the suppliers to the network has issued that for some reason proved incompatible or failed – and I guess now they're going to be looking at whether they can reverse that out again or fix that."
Steve Ranger, UK editor of Tech Republic, a technology news site, said it was a "very large outage which is undoubtedly incredibly annoying for the customers and very embarrassing for O2 as well".
"The unfortunate truth is that these systems are incredibly complicated and if you… change something, you can cause all sorts of problems down the line," he added.
Tom Morrod at market research firm IHS Markit said data was increasingly important to consumers, with half of UK mobile users prioritising internet connectivity above calls and texts.
"As well as the inconvenience to consumers and the associated frustration, having a major network out of action creates productivity challenges for businesses. Many businesses will have employees commuting or out in the field that have lost work time today," he said.
One O2 customers, Allison Rose-Mannall, from Norwich, is an insulin-dependent diabetic who relies on her mobile and is unable to get to a landline.
"I'm disabled … I'm in a wheelchair," she told the BBC. "So having no data but also no calls as well means I can't contact anyone if I have a fall or if I need anything."
Lynsey Greaves runs a company in Doncaster providing home visits to the elderly and vulnerable. Her 130 staff all rely on O2 phones to access rotas, schedules, names and addresses. Since 05:30 she's been calling in extra office staff to give out the information for each visit over the phone.
"There are nine of us trying to sort it now," she said. "It's been a nightmare."
Luke Stagg is trying to run a plumbing business via his phone, but he can't get through to customers or use his sat nav.
"That's a whole day wasted," he says. "I'll be seeking to recoup my losses, especially as a business customer."
But a few people have actually welcomed the forced screen-break.
"Remember the not so distant past, when people weren't wandering around like zombies, glued to social media?" says O2 customer David Flanders. "Remember striking up a chat with the stranger next to you on the train or the bus?
"I am certainly not going to lose any sleep over losing my data for a while."
Can I claim compensation?
Customers of O2 will be able to claim for any out-of-pocket expenses that resulted from being without their phone, according to consumer expert Helen Dewdney.
Ms Dewdney, who writes a blog called The Complaining Cow, said users should be able to claim under the Consumer Rights Act 2015, which states that services should be carried out with reasonable skill and care.
That means customers would be able to claim a refund for what they would pay on a contract for the time they were without the use of the phone. They could also claim consequential loss due to breach of contract, for example, if they incurred bank charges because they were unable to move their money, or the cost of having to use a payphone.
This must be a genuine loss which can be proved with evidence. So, a taxi driver might be able to prove they lost out on fares owing to the shutdown, but other workers trying to claim a lost day of employment would struggle.
Ms Dewdney suggested phone users calculated their losses, and wrote to O2 with the evidence. They should state what they wanted as redress and mention the Consumer Rights Act 2015.
If they felt the response was unsatisfactory, it could be referred to Ombudsman Services: Communications of which O2 is a member. The customer would need to ask for a deadlock letter or wait eight weeks from the beginning of the complaint.