Train strikes are causing fresh disruption on Saturday in the third walkout this week.
Mick Whelan, of rail drivers' union Aslef, apologised for any inconvenience and said they did not want to "disturb ordinary working people".
But he warned the strikes could continue for years until a resolution is found.
The government said it had offered a fair and reasonable pay offer and urged union leaders to "do the right thing".
This is the third strike to affect train services this week and comes on the same day as the FA Cup Final at Wembley, a Beyoncé concert at the Tottenham Hotspur stadium, a Test match at Lord's Cricket Ground and the Epsom Derby.
Aslef, which represents train drivers, is calling for a better pay offer and wants the government to take action.
General secretary Mr Whelan said: "I don't want to be doing this and my members do not want to be losing money and do not want to be on strike."
Speaking to the BBC, he added: "I haven't seen the transport minister since before Christmas."
Train drivers have not had a pay rise for four years, he said, adding that the union had only received two "non-offers" from the government.
Asked if there was a light at the end of the tunnel, he said there was none "at this moment in time".
Speaking to Sky News earlier, Mr Whelan said the strikes could continue for years, adding: "We are in this if it takes us four years, five years, whatever it is."
While the government reached a pay deal with signal workers and maintenance staff employed by Network Rail in March, it remains in dispute with drivers, who are largely represented by Aslef, and other rail workers who are members of the RMT.
The Aslef union has rejected a two-year offer which would see drivers get a backdated pay rise of 4% for 2022 and a 4% increase this year.
Aslef members also walked out on Wednesday and the RMT union held strike action on Friday.
Some tourists from the US told the BBC on Saturday that they had been forced to rent a car for $1,300 to get from London to Manchester to catch a flight.
Taylor Haney said the family of five had brought train tickets, but when they arrived at the station the platform was closed.
"I wasn't really expecting to spend that kind of money, or drive again, but we'll make it," he said.
Mr Haney, from Tennessee, said it was "tough" changing travel arrangements with three children and he wished the process had been "a bit smoother".
Although the affected train operators are in England, some of their services run into Scotland and Wales so journeys to those nations have also been disrupted.
The Rail Delivery Group said passengers with advance tickets could be refunded fee-free if the train that the ticket was booked for was cancelled, delayed or rescheduled.
A spokesperson for the Department for Transport said union leaders had coordinated the strikes to disrupt passengers in a week of major events.
"Not content with impacting the hundreds of thousands of people who have looked forward to these events all year round, unions are also targeting their own members' pockets by forcing them to miss out on pay every time they strike," it added.
The government urged union leaders to put the "fair and reasonable" pay offer to their members.
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Train strikes hit FA Cup final and Beyoncé concert
