Easyjet chief executive Andy Harrison said his company would be seeking £50m compensation from the government, while British Airways chief executive Willie Walsh said it would take weeks to get back to normal.
Ryanair has said it will defy EU regulations and warned customers it will only reimburse customers their air fare and no additional expenses.
More than 95,000 flights were cancelled across Europe over the past six days, with only a handful of flights taking off and landing at UK airports.
The eruption of the Icelandic volcano Eyjafjallajokull on 14 April sent vast amounts of ash into the atmosphere and posed a threat to aircraft jet engines.
Scientists say the volcano is still erupting but the ash plume is shrinking, although it remains changeable.
Nats said a dense concentration of volcanic ash would remain over the north of Scotland until 0100 BST on Thursday.
Passengers toast lucky tickets
Restrictions were lifted after the Civil Aviation Authority (CAA) said safety tests showed plane engines had "increased tolerance levels in low ash density areas".
It has set down new requirements for airlines such as conducting risk assessments and inspecting aircraft for ash damage before and after each flight.
CAA chief executive Andrew Haines said he would not apologise for the delay in opening UK airspace as safety was the first priority.
Prime Minister Gordon Brown also defended the time taken to reopen UK airspace, saying decisions had been based on "scientific advice".
Lord Adonis said the ban
was lifted after a "robust safety assessment" based on observational data and test flights. He denied the decision to reopen the airspace was the result of pressure from the airline industry.
Conservative leader David Cameron called for a "rapid inquiry" into the handling of the crisis and said there had been "muddle and confusion" over the information people were given.
Liberal Democrat leader Nick Clegg, whose three sons were stranded in Spain, said a "post-mortem" of how the situation was dealt with would be needed.
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