American hikers Josh Fattal and Shane Bauer have taken off from Tehran and are on their way to Oman after being released from an Iranian prison Wednesday, a Swiss diplomat and their lawyer told CNN.

Livia Leu Agosti, the Swiss ambassador to Tehran, said the pair were bound for the Omani capital, Muscat.

The men’s families, as well as fellow hiker Sarah Shourd — who was arrested with them but freed last year on medical grounds — were heading to Oman’s airport to meet them, a spokeswoman said. Shourd is now Bauer’s fiancee.

The families expressed their joy, relief and gratitude at the pair’s release.

“Today can only be described as the best day of our lives,” their statement said. “We have waited for nearly 26 months for this moment and the joy and relief we feel at Shane and Josh’s long-awaited freedom knows no bounds.

“We now all want nothing more than to wrap Shane and Josh in our arms, catch up on two lost years and make a new beginning, for them and for all of us.”

Oman’s envoy to Tehran, Salem al Ismaily, earlier said in a statement the pair were in the custody of the Omani government and would spend a couple of days in Muscat “before heading home.”

The two Americans were released on bail of $500,000 each and their sentences for spying convictions were commuted, Iran’s judiciary said, according to government-run Press TV.

The lawyer for Fattal and Bauer, Masoud Shafiee, told CNN the $1 million bail had been paid by the Omani government.

The pair arrived at Tehran’s Mehrabad International Airport late Wednesday, in a convoy of diplomatic cars with tinted windows which headed for a section of the airport normally used only by Iranian government officials.

The convoy had earlier left the country’s notorious Evin Prison through the front gates, accompanied by a police escort, but Bauer and Fattal were not visible.

Swiss and Omani officials had waited outside the prison to receive the Americans. Switzerland represents United States interests in Iran because there is no American embassy there, and Oman has acted as a broker between Washington and Tehran in the past.

The two men’s release comes a day before Iranian President Mahmoud Ahmadinejad is due to speak at the United Nations General Assembly in New York.

Asked at the United Nations about the news, Ahmadinejad’s chief of staff, Esfandiar Mashaei, said: “Yes, we were expecting their release, us and the president.”

High-profile American Muslims including boxing legend Muhammad Ali had called for their release, and a high-profile delegation of American Christian and Muslim religious leaders met Ahmadinejad in Iran last week to plead for their freedom.

“We were very happy to learn about their release today,” said Nihad Awad, national executive director of the Council on American Islamic Relations, who was part of the religious delegation to Tehran.

“We are extremely happy for the hikers, their families and the country.”

Bauer and Fattal, both 29, were convicted last month of entering Iran illegally and spying for the United States, and each was sentenced to eight years in prison.

Their attorney, Shafiee, went into the prison earlier Wednesday with paperwork to show that bail had been paid for each of them. A judge had signed the bail papers that morning after several days of delays.

Fattal and Bauer were arrested along with Shourd in July 2009 after apparently straying over an unmarked border between Iraqi Kurdistan and Iran.

Bauer and Shourd told their families in May last year that they had gotten engaged in prison and planned to get married after their release. Fattal plans to be best man at the wedding, the hikers’ relatives said in a statement at the time.

Shourd, in an interview with CNN a year ago, showed the improvised engagement ring that Bauer had woven from thread pulled from his shirt.

Shourd said Bauer had asked her to marry him while they were imprisoned so that they could have something to sustain them through their days in Evin, and give them hope for their future together.

Shourd was freed almost exactly a year ago on medical grounds. Her release came a week before Ahmadinejad addressed the U.N. last year.

One analyst said the timing last year was no coincidence.

“I think President Ahmadinejad really wanted to use this as a way of building up a store of goodwill just before he comes to New York,” Columbia University Prof. Gary Sick said last year after Shourd came home.

Oman helped secure the release of Shourd, posting her bail last September, a senior Obama administration official said at the time.

The Americans say they accidentally crossed into Iran when they veered off a dirt road while hiking near a tourist site in the Kurdish region of northern Iraq. They denied the charges and appealed the sentence while serving time in prison.

Ahmadinejad said last week that the release of Fattal and Bauer was imminent, setting off a roller coaster ride of expectations.

The judiciary shot back that only it could make decisions about their release.

Shafiee then announced that all the paperwork had been filed for them to be freed on bail, but their release was delayed by the lack of a judge’s signature on bail paperwork.

An Omani official flew to Iran on September 14 to help work on any negotiation, a Western diplomat told CNN at the time.

CNN’s Elise Labott, Shirzad Bozorgmehr, Mitra Mobasherat and Susan Candiotti contributed to this report.