Hungary's new prime minister, Péter Magyar, is set to be sworn in, almost a month after he steered his Tisza party to a landslide victory, sweeping away 16 years of rule by Viktor Orbán.

Tisza holds 141 seats out of 199 in the new parliament – up from zero, a result of the party being founded just two years ago.

A big "celebration of freedom and democracy" is planned on Saturday in front of parliament in Budapest, along the shores of the Danube. Magyar has told Hungarians to step through the "gateway of regime change".

Orbán's Fidesz party crumbled from 135 to 52 seats, and shows signs of imploding.

He and other key party figures have decided not to take their seats in parliament, and their political future is uncertain, beyond a vague commitment to "rebuild the national side".

Each day brings new revelations or allegations of corruption against a party which has governed Hungary almost unchallenged since 2010. Magyar has promised a "change of system" as well as a change of government.

It is not known whether Orbán will attend Saturday's opening session of parliament, even as a simple guest.

"The main priority is to set up the government… on the ruins of the previous one," Zoltán Tarr, incoming Minister for Social Relations and Culture, told the BBC.

"We are ready to face a very grim economic situation. But at the moment, we just don't know the severity."

A spending spree initiated by the Orbán government in the past eight months came on top of years in which state contracts and funds were channelled to business circles close to Fidesz.

The budget deficit has already swollen close to the planned target for the whole year.

The incoming government is at pains to show that it is morally stronger than Fidesz.

One prominent businessman, György Wáberer, who switched from Fidesz to Tisza a week before the election, told a journalist he had donated £242,000 (€280,000, $331,000) to Tisza.

Magyar promptly returned the money to him.

When Magyar's brother-in-law, Márton Melléthei-Barna, was named justice minister, the new government was bitterly criticised on social media.

On Thursday evening, Melléthei-Barna announced that he was withdrawing his candidacy for the post, "to ensure that not even the slightest shadow is cast on the transition".

Incoming Tisza ministers say there will be no revenge against the outgoing government, but those guilty of financial crimes will be held accountable. A new "office to recover stolen assets" will be set up.

"I don't think that we should talk about a guillotine," said Tarr, in response to calls for those responsible for siphoning off the national wealth to go on trial.

"We are talking about investigations and actions which are totally in line with the rule of law. Interestingly enough, the current chief prosecutor, and the police, have started certain investigations which they did not start before the election. They are questioning people."

The small number of prosecutions of prominent figures in Hungary in the past "is turning into a steady flow", a source close to the prosecutor's office told the BBC, "not because we didn't want to prosecute before, but because the police and the tax office were reluctant to gather evidence".

"What has changed is that people are now coming forward. So a lot more evidence is suddenly available," the source added.

One target of police investigations is the media empire of Gyula Balásy, which won millions in government contracts over the past decade, and ran Fidesz campaigns hostile to migrants, as well as "enemies", who ranged from billionaire businessman and philanthropist George Soros, to Ukrainian leader Volodymyr Zelensky and Magyar himself.

In a tearful interview last week, Balásy said he wanted to hand over his companies and investments to the state, though he denied any wrongdoing.

The accounts of some of his companies have been frozen.

Another target of investigation is Hungary's National Cultural Fund, and within it an Urban Civil Fund – with a budget of £57.2m (€64.9m, $76.7m). Allegations that it may have channelled money to Fidesz candidates are also being investigated.

The biggest challenge facing the new Magyar government is the urgent need to release €17bn in various EU funds, withheld by the European Commission from the Orbán government.

Last week, Commission sources in Brussels suggested that some of the Hungarian money may be lost.

Other problems include the need to find a common position on an EU migrant pact, which was bitterly opposed by the Orbán government.

The pact is due to come into full force on 12 June, but Hungary is still being fined €1m a day for non-compliance with rulings of the European Court of Justice (CJEU) over its treatment of migrants.

Recent surveys also suggest that Tisza party voters, like Fidesz voters before them, are worried about irregular migration.

They are also wary of Ukraine's desire to join the EU. Magyar, like Orbán, says Hungary still needs Russian oil and gas supplies, at least in the short term.

But Tarr is optimistic. He sees a willing partner in the EU and is not concerned that young people who danced through the night on 12 April will become disillusioned, after they return to continue celebrating this weekend.

"I'm not worried, I'm excited… We are serving the country. We are serving the people. We are not here to rule. We are here to serve. We are here to fulfil a mandate."