Who had the shortest time in going from imprisonment to head of state within democratic processes?

And we got a new winner! Sadyr Japarov was released/liberated from prison on October 6, 2020 and became president of Kyrgyzstan on January 10, 2021. AlJazeera:Who is Kyrgyzstan's new prime minister?

Václav Havel was jailed on 16. January 1989 and released in May on parole. He was shortly jailed also in October 1989. This paid congratulation announcement was published by Rudé právo when he was in jail. The name is just a cover-name from his theatre plays.

He was elected the president on 29. December 1989 still by the Communist parliament. It was not a democratic parliament, but the process followed the constitution of the comunist state. He was re-elected by a democratically elected parliament on 5. July 1990.

The Czech Wikipedia describes it this way:

V pondělí 16. ledna 1989, na počátku série demonstrací během Palachova týdne, byl zatčen a v únoru odsouzen k devíti měsícům vězení. Po odvolání mu byl trest snížen na osm měsíců, podmíněně propuštěn byl Havel již v květnu. V červnu 1989 stál u zrodu petice Několik vět. V září se stal mluvčím Charty 77 za uvězněného Alexandra Vondru. Naposledy zatčen byl v říjnu 1989, brzy nato však byl propuštěn na svobodu.[28]

which means:

On Monday 16. January 1989, at the beginning of a series of demonstrations during the Palach's week, he was detained and in February he was sentenced to nine months of prison. After appeal the sentence was reduced to eight months and he was released conditionally in May. In June 1989 he was at the inception of the "A Few Sentences" petition. In September he became the spokesperson of Charter 77 in place of the imprisoned Alexandr Vondra. For the last time, he was detained in October 1989, but he was soon released.[28]

And later:

Když padla komunistická vláda Ladislava Adamce a prezident Gustáv Husák přislíbil svou abdikaci, bylo nutno hledat nového prezidenta republiky. ... Kombinací všelidového nátlaku a vyjednávání v poslaneckých kruzích, ..., se podařilo přesvědčit i komunistické poslance k volbě jejich nedávného úhlavního nepřítele. Tak byl Václav Havel 29. prosince 1989 ve Vladislavském sále Pražského hradu jednomyslně zvolen prezidentem Československé socialistické republiky, což byl stále ještě oficiální název státu.

In English:

When the communist government of Ladislav Adamec fell and president Gustáv Husák promised to abdicate, it was necessary to find the new president. ... Through a combination of all-people pressure and negotiation among the members of the parliament, ..., they succeeded to convince even the communist PMs to vote for their recent archenemy. That way Václav Havel was unanimously elected on 29. December 1989 in the Ladislaus Hall of the Prague Castle to be the president of the Czechoslovak socialist republic, which was still the oficial name of the state.

I think the answer to your question is former Egyptian president Morsi, he escaped prison on January 30th 2011 and was sworn in as president on June 30th 2012.

Józef Piłsudski was released from German prison on 8 November 1918 and became Chief of State 73 days later, on 20 January 1919.

Some timeline in between:

• he was appointed Commander in Chief of Polish forces just 3 days after being released, on 11 November 1918 (or just one day after, depending on how you count; he left prison on 8th, but was then escorted to Warsaw, and left by the German soldiers on 10 November),

• on 22 November, he was appointed a Provisional Chief of State. This was, however, not a democratic process,

• on 20 January 1919 he resigned, and was immediately re-instated as Chief of State by the elected parliament, so albeit not elected directly, he became a head of state in a democratic process.

Olusegun Obasanjo, President of Nigeria from 1999 to 2007 is a possible candidate.

He was imprisoned in June 1995 on charges of being involved in plotting a coup and was released in June 1998. Although a general, he then ran as a civilian for president in February 1999 and won the election with almost 63% of the vote as the candidate of the People's Democratic Party.

Obasanjo was convicted based on evidence obtained through torture and was widely perceived to be a political prisoner as he had criticized the Sani Abacha regime of human rights abuses. His sentence was reduced from life to 15 years "after pressure from friends abroad, including South Africa's Nelson Mandela, former US President Jimmy Carter and former German Chancellor Helmut Schmidt." The imprisonment of Obasanjo and others by the regime of Sani Abacha was condemned internationally. When Sani Abacha died on the 8th of June 1998, his successor Abdulsalami Abubakar ordered that Obasanjo and eight other political prisoners be released. Obasanjo was freed about a week after Abacha's death and elections were held in February 1999, overseen by the Independent National Electoral Commission.

Obasanjo has the distinction of being Head of State both before and after being imprisoned. His first stint was from 1976 to 1979 as a military ruler following the assassination of General Murtala Mohammed. He voluntarily left office in 1979, handing over to the democratically elected Shehu Shagari. This, combined with his military contacts and the fact that he was widely perceived to have been imprisoned for political reasons, meant that the electoral commission had no problem with a former prisoner running in the election.

When Obasanjo was last in office he gained international respect through his efforts to end white minority rule in South Africa and Zimbabwe, supporting neighbouring states such as Angola and Mozambique. After relinquishing power, Obasanjo retired to his second home at Otta, outside Abeokuta, where he enjoyed his image as a man of the people....

Obasanjo's argument that his own military background makes him uniquely qualified to keep the Army under control has carried more weight among the electorate than fears that he will not be his own man. He also won widespread credibility when he voluntarily handed over power to a civilian government in 1979.

Source: Obasanjo Biography (University of Pennsylvania, African Studies Center)

Although there were some reported irregularities and the result of the election was contested by Obasanjo's rival, former finance minister Olu Falae, the Nigerian courts rejected requests to have the result annulled. Obasanjo was elected for a 2nd term in 2003.

You may also want to consider Éamon de Valera who escaped from prison in February 1919 and became President of Dáil Éireann in April 1919 (2nd Minsitry) under it's unilateral declaration of independence. However, the Irish Republic of 1919-22 was not recognized by the major powers of the time.

De Valera had been elected Sinn Fein MP for East Clare in July 1917 after being released in June 1917 (he had been imprisoned in 1916 following the Easter Rising). He was rearrested in May 1918 and then escaped from Lincoln jail in February 1919. During his imprisonment, he was also elected for a second constituency in the December 1918 General Election. Unsurprisingly, as far as most Irish voters were concerned, his prison record was no handicap.