The Greenpeace "Arctic 30" could be home by Christmas, and the two jailed members of the punk group Pussy Riot should be released from jail in the coming days, after a wide-ranging amnesty law was passed by the Russian parliamenton Wednesday .

The Pussy Riot pair are serving a two-year sentence for hooliganism motivated by religious hatred, while the Greenpeace activists are charged with hooliganism and are currently on bail awaiting trial in St Petersburg.

The amnesty, backed by Russia's president, Vladimir Putin, is timed to coincide with the 20th anniversary of Russia's constitution. It mainly concerns first-time offenders, minors and women with small children.

An amendment on Wednesday extended the amnesty to suspects in cases of hooliganism, which includes the Arctic 30, who were arrested aboard the Greenpeace ship Arctic Sunrise in September.

They were bailed by courts in St Petersburg last month but still faced trial and potential jail sentences of up to seven years.

Iain Rogers, one of the Greenpeace activists, in a cage at a bail hearing in Murmansk in October. Photograph: Igor Podgorny/Greenpeace/SWNS.com

The Greenpeace activists expressed relief, though Arctic Sunrise captain Peter Willcox said: "There is no amnesty for the Arctic."

He added: "I might soon be going home to my family, but I should never have been charged and jailed in the first place."

Kieron Bryan, a freelance journalist and one of six Britons among the 30 Greenpeace detainees, said that with all the uncertainty about whether or not they would be included in the amnesty, the past week had been hard to cope with: "We've all been feeling the emotional strain this week," he said from St Petersburg.

"We'd heard rumours about the possibility of this at every stage of the process, but there have been so many rumours and false hopes that I never really believed it. For every positive thing that's happened there has been a setback, and a feeling that we might be here for a very long time and go to jail."

Greenpeace communications officer Alexandra Harris said: "We are relieved we are coming home but we don't know when. It is quite a strange feeling.

"Our amnesty will be signed off tomorrow and then the investigators [in the case] will have to approve it and then we have to wait for visas. It could take weeks or we could be home for the weekend. That would be amazing if we could be home for Christmas."

Harris, who works for Greenpeace in Australia, said she was looking forward to spending time with her family in Devon and a long walk on Dartmoor.

Mikhail Khodorkovsky reads documents behind a glass wall during a court session in Moscow in June 2010. Photograph: Sergei Karpukhin/Reuters

She added: "It's strange that we are being forgiven for a crime we didn't commit, and I keep thinking about my Russian friends. I always imagined we'd all be together in this moment and let go under the same circumstances. We've been a group this whole time and I thought we would be sharing this moment – but the amnesty doesn't mean the same for all of us."

The Duma, Russia's parliament, voted 446-0 in favour of the bill in its third and final reading on Wednesday . Once it is printed in the state newspaper, Rossiiskaya Gazeta, probably on Thursday , it will then become law.

The amnesty does not contain names so there is still uncertainly about how exactly it will be applied, but it appears that the decision has been taken on high to improve Russia's image ahead of the 2014 Winter Olympics in Sochi, which begin in less than two months.

Theoretically, once the law is published, the next step for the Greenpeace activists will be to apply to Russian investigators to end the case against them, and then they will need to apply for exit visas. As they were brought to Russia forcefully, they do not have proper entrance visas. It is a bureaucratic hurdle that could take some weeks to sort out, but it is also possible that now a decision has been taken, efforts will be made to arrange the paperwork quickly.

Bryan said it was unclear whether or not their visas could be ready in the next week: "It could be in time for Christmas, but whatever happens it's a massive weight off my shoulders, and for my parents to know that their son is coming home. That's the main thing, even if I end up missing Christmas."

Sue Turner, the mother of Iain Rogers, one of the six British nationals among the Arctic 30, said: "Until I have heard it officially, I can't quite believe it.

"I am waiting to know whether they have to wait for exit visas or if they are put on a plane straight away. I am so excited, I just can't take it in. It is a great Christmas present for me and the family and all Iain's friends."

The other high-profile beneficiaries of the amnesty are Nadezhda Tolokonnikova and Maria Alyokhina of the punk group Pussy Riot, who are serving two-year sentences for staging an impromptu punk performance in Moscow's main cathedral early last year.

Petya Verzilov, Tolokonnikova's husband, said he believed an order had been given to speed up the process. Although technically releases could take up to six months to be processed from the day the law is published, officials at both prisons have indicated they are ready to release the Pussy Riot duo as soon as soon as the law is passed, he said.

Alyokhina is serving her time in a prison in the city of Nizhny Novgorod, while Tolokonnikova was recently moved from Mordovia, a region known for its Soviet-era gulags, to the Siberian city of Krasnoyarsk.

She has said that the conditions are incomparably better than in Mordovia, from where she published a long open letter detailing slave-like conditions of forced labour and cruel punishments.

Verzilov indicated he expected them to be released as soon as Thursday , and was planning to fly to Krasnoyarsk late night.

"They are slightly sceptical of course," Verzilov told the Guardian. "When you're living in these conditions it's hard to think about the Duma passing some bill, and it seems like it could never happen, so it's a big surprise for them that it does actually seem to be happening."

Tolokonnikova and Alyokhina, who were both sentenced to two years in jail last summer, were due to be released in early March, as their sentence included time served since their arrest. A third member of the group, Ekaterina Samutsevich, was freed on appeal shortly after the trial concluded.

Verzilov said that on their release, the pair plan to launch a major new project related to the Russian prison system, though he declined to give details.

While Greenpeace and Pussy Riot are celebrating, there are notable exceptions to the amnesty. It does not cover Russia's former richest man, Mikhail Khodorkovsky, who has been found guilty of economic crimes at two separate trials, and neither does it include many of the people on trial for disturbances at a rally the day before Putin was inaugurated last year.

Some of those accused will receive an amnesty, but more than half of the 26 involved will not, including Mikhail Kosenko, who was sentenced to indefinite psychiatric treatment in October, despite the court hearing he was not involved in the violence and his lawyers saying he was of sound mind.

"Theoretically I can apply tomorrow to be released from court proceedings and go abroad for New Year," said Maria Baronova, one of those on trial who does fall under the amnesty. "But this is Russia after all. Maybe the prosecutors will think something up overnight and it won't happen."