Short answers, pulled from the story.
Władysław Raczkiewicz was appointed as the successor under Article 24 of the Constitution adopted in April 1935. He took his constitutional oath at the Polish Embassy in Paris after Mościcki did not resign until the 29th or the 30th of September 1939.
The government established its initial base in France during 1939 and 1940, first operating from Paris before moving to Angers. Most of the Polish Navy managed to escape to Britain while tens of thousands of soldiers fled through Hungary and Romania or crossed the Baltic Sea to continue fighting in France.
Stalin subsequently severed all diplomatic relations with the Polish government following their refusal to accept the Soviet explanation for the discovery of mass graves containing 10,000 Polish officers near Smolensk Russia who had been captured in 1939. This breach occurred alongside an air crash that killed Prime Minister Władysław Sikorski at Gibraltar in July 1943.
Foreign Minister Count Edward Raczynski sent a twenty-one point summary to Governments of the United Nations regarding killing of Jews in Poland. He confirmed that German authorities aimed with systematic deliberation at total extermination of Jewish population in Poland.
On the 5th of July 1945 France followed by United States and United Kingdom withdrew recognition of the government-in-exile after formation of Provisional Government of National Unity. The last country to withdraw diplomatic recognition was Holy See Vatican City on the 19th of October 1972 though diplomatic privileges had been withdrawn earlier by Vatican Secretary of State Domenico Tardini in 1959.
Last president Ryszard Kaczorowski passed responsibilities and insignia to Lech Wałęsa first non-Communist president since war on the 22nd of December 1990 at Royal Castle in Warsaw. Liquidation of London-based apparatus declared accomplished on the 31st of December 1991 ending decades of exile operations.