The impending conclave reminds me that around the house there are four accounts of papal elections, covering six centuries and then some.
The oldest and most intimate is by Pius II, recounting his own election in 1458. Florence Gragg's translation of his memoirs, Secret Memoirs of a Renaissance Pope: The Commentaries of Aeneas Sylvius Piccolomi, Pius II, is out of print in the older paperback edition, but Amazon seems to have it available at a range of prices. Harvard University Press provides a bilingual edition of the Commentaries; the English is Gragg's, with some revisions.
Chateaubriand was ambassador to Rome during the election of Pius VIII in 1829. What he saw of it, he tells in Mémoires d'Outre-Tombe largely by official letters and letters to Mme. Recamier. Chateubriand obtained, he said, the journal of the conclave; this he translated into French, redacting it so that the sources could not be identified, and sent to the foreign minister, with his comments on it, enjoining strict secrecy. The Italian original he burned. He was happy with the results of the conclave, for the pope, formerly Cardinal Castiglione, was regarded as favorable to France, though his secretary of state, Cardinal Albani, was known to favor Austria.
Stendahl gives a circumstantial eyewitness account of the events of 1829, in the first person, though he was not in Rome. His Roman Journal is written mostly from the notes of a cousin who had the material but not the skill in writing. If you wish to read a dramatic account of Cardinal Albani stepping out to the balcony to say "Annuntio vobis gaudium magnum, papam habemus", you will have to consult Stendahl, who was in Paris, rather than Chateaubriand, who was in Rome.
The American journalist Murray Kempton was in Rome during the election of John Paul I. His account is collected in Rebellions, Perversities, and Main Events. It is certainly not the best essay in the collection; quite possibly I'd have forgotten it but for its mention of Stendahl. But the essay and the book are both well worth reading.