Travel disruption caused by the discovery of cracks in some high-speed trains is expected to continue this week.
Robert Nisbet, of the Rail Delivery Group, said initial inspections of 183 trains taken out of service on Saturday should conclude by the end of Sunday.
But he warned that disruption was expected to "carry on for a few days".
GWR advised against travel on Monday, while LNER urged people with bookings to check before they travelled.
Hitachi Class 800 trains used by the firms were taken out of service for safety inspections.
Andrew Barr, CEO of Hitachi Rail, apologised to passengers and said the trains were withdrawn because of cracks found on the metal that linked the train's body with the underside of the train, known as the bogie.
On Sunday evening, the UK government asked Hitachi to set out a comprehensive plan to identify the extent of the cracking and outline a longer term repair strategy.
While this sort of problem doesn't affect the structural integrity of the train - so it's not a danger to passengers - I'm told the risk is more about the possibility of something falling off.
GWR is the most affected network. As the first operator in the country to introduce the Hitachi 800 in 2017, its trains are older.
Its fleet of 93 is also much bigger than other operators', meaning more trains to check and fix.
Some are already back in service - and we're told inspections should be done by the end of Sunday.
And Hitachi Rail's boss told me the company would periodically reinspect the trains while a permanent solution is found.
Disruption on long distance routes will go on for at least the next few days. That's partly because some trains and crew won't be in the right place.
Also, some carriages won't be able to go straight back into use because they need repairing. We don't yet know precisely how many yet. It's possible some which aren't yet showing a crack might also need remedial work.
Mr Nisbet, whose organisation represents train operators and Network Rail, also apologised for the disruption.
He told BBC Breakfast that rail firms were running as many services as possible, including rail replacement services, putting on extra staff at affected stations, relaxing ticket acceptance rules and offering compensation for delays.
He said it was "gratifying to see the speed" at which initial inspections were taking place but said it was impossible to say how long the checks would take. "We don't want to rush it," he said.
Mr Nisbet said the cracks - measuring millimetres - "didn't pose any particular danger to passengers" but had the potential to develop if left untreated.
The Department for Transport said independent safety regulator the Office of Rail and Road (ORR) will be closely involved in overseeing Hitachi's plans for safety inspections for the Class 800 fleet.
Rail Minister Chris Heaton Harris said: "I expect operators to explore all options for replacement services to help people complete their journeys." He thanked passengers for their patience in the meantime.
Instagram photographer Anna Celac hoped to travel from Cardiff to London Paddington to see a friend she hadn't seen since last year.
When she bought her ticket on Saturday night, she said the website she booked with didn't report any disruption.
She says she was disappointed when she had to cancel her trip, as the replacement bus service would have left her with too little time in the capital.
Only around 10% of Great Western Railway's long-distance services ran on Sunday, with a similar service expected on Monday, the operator said.
GWR said customers for long-distance services should not travel on Monday and refunds would be issued.
Local services have been running, but they were expected to be very busy and rail replacement services were limited.
The company earlier said cracks were detected on "more than one" Hitachi 800 train, so all 93 in its fleet were being inspected as a precaution.
It comes after six of the trains were taken out of service by GWR last month after hairline cracks were found in the suspension system during routine maintenance.
London North Eastern Railway apologised to customers, saying it was experiencing "significant disruption" due to the checks.
It is running a limited service on the East Coast Main Line, between Edinburgh, Newcastle, York and London, and will be unable to run trains between Edinburgh and Inverness/Aberdeen until Saturday 15 May at the earliest.
LNER added that some services would also be cancelled on Monday - and passengers should check before setting off to the station.
Hull Trains said checks on its trains had been completed and services would resume as normal.
But the operator added it expected services to be "extremely busy" and could not guarantee that social distancing guidelines would be adhered to. Refunds were being offered for its Sunday morning services, according to its website.
TransPennine Express said it was now running a normal service, but advised customers travelling between York and Newcastle to check for any knock-on impact before travelling.
LNER passengers affected by cancellations can use their tickets on the following operators' services instead:
- ScotRail between Aberdeen and Edinburgh
- Avanti West Coast between London Euston and Manchester for customers with an open ticket
- TransPennine Express between Manchester, Leeds and York
- East Midlands Railway between London St Pancras International and Sheffield
- Northern services between Sheffield, Leeds and York and via any other "reasonable" route
- Thameslink services between London, Stevenage and Peterborough
- CrossCountry between Leeds, York and Edinburgh
Rail replacement coaches are operating at York, Newcastle and Edinburgh, and also between Doncaster and Peterborough.
LNER says affected customers' tickets will be valid up to and including Sunday 16 May, though new reservations will need to be made.
GWR passengers can use their tickets on the following services:
- Avanti West Coast trains between Birmingham New Street and London Euston
- London Northwestern Railway trains between Birmingham New Street and London Euston
- West Midlands Railway trains between Hereford/Worcester and Birmingham New Street
- Chiltern Railways via any "reasonable" route
- CrossCountry via any "reasonable" route
The Hitachi Class 800 train
- The Class 800 is electric - but almost half have a diesel engine enabling them to operate on lines that have not been electrified
- Formal orders for the trains were made in 2012 and 2013 at a cost of around £5.7bn for a 27.5 year programme, including maintenance
- The Class 800 began regular service for Great Western Railway in October 2017 - but an embarrassing start saw air conditioning units leak on to passengers on the first Class 800 journey from London to Bristol
- In May 2019, they began service on the East Coast Main Line, with operator LNER branding the train Azuma - or "east" in Japanese
- They were hailed by operators as helping to increase capacity and reduce journey times
- The trains were assembled at Hitachi's plant in Newton Aycliffe in County Durham
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