There was indeed a ceremony of accession performed by Qing emperors known as 登極 dengji (lit. 'ascending [to the] ultimate'), the full instructions for which are laid out in Book 23 of the 大清會典 Daqing Huidian, the statute books compiled under the auspices of the Qianlong Emperor. This ceremony might be better described as an enthronement rather than a coronation, given that the latter word's etymology implies the bestowment of a crown (corona), an object which Qing rulers did not use. The ceremony took place in the 太和殿 Taihedian('Hall of Supreme Harmony') in the Forbidden City, which housed the 龍椅 longyi ('Dragon Throne') on which the emperor would be seated. For the occasion, a series of tables would be placed in the chamber: one for the imperial seal, placed directly in front of the throne; one for the formal petition for the emperor's accession, placed to the south of the eastern pillar; one for the edict announcing the accession, placed to the north of the same pillar; and one for the writing instruments, placed by the western pillar. The imperial vestments would be placed outside the hall, while the carriage would stop outside the Taihe gate. The palace musicians would set up along the east and west sides of the causeway into the hall, and the 天安門 Tiananmen ('Gate of Heavenly Peace') would be prepared for the reading of the imperial edict. The ceremony's attendees would arrive before the emperor in the open space just outside the hall, with members of the imperial clan on the steps up to the hall, and other civil and military officials behind them.

While the specific ritual of dengji took place more or less entirely within the bounds of the Taihedian, in practice the accession ceremony was performed immediately after a series of preparatory rituals. The first two were mourning rituals for the new emperor's predecessor: the first would be performed in mourning clothes in front of the dead emperor's coffin, to which his successor performed the 叩頭 koutou (kowtow) (which involved kneeling, knocking the forehead against the ground three times, standing, and repeating the process until there had been three kneelings and nine knockings); the second would be performed in imperial dress in front of the Empress Dowager (also in ritual dress), and again involved performing a koutou. After these rituals would come a rite of homage: at an auspicious moment decided by the head of the Board of Astronomy, the emperor would leave the inner palace for a secondary audience chamber, the 中和殿 Zhonghedian ('Hall of Central Harmony'), where a group of senior officers would perform the koutou, followed by the Board of Rites calling upon the new emperor to ascend the throne. Only then would the emperor, officials and attendants process to the Taihedian. Beforehand, there would be a set of sacrifices performed by lesser officials at four altars in the palace: the altars to Heaven, Earth, the Imperial Throne, and the Gods of Agriculture.

In the Taiheidian, the emperor would advance to the imperial throne and sit facing south, a moment marked by the ringing of bells and the beating of drums by the palace musicians, and the cracking of a whip by the commander of the imperial bodyguard. This would be followed by a koutou of those in attendance, and then the edict of accession would be sealed. The emperor would retire and resume the funeral of his predecessor, while the edict of accession would be brought out to the Tiananmen and read out by a minister, after which those outside the gate would perform a koutou. The edict would then be passed to a member of the Board of Rites, who would read it before a smaller group of officials at the 乾清門 Qianqingmen ('Gate of Heavenly Purity'), who would again perform a koutou after the reading. This edict would include the announcement of the imperial reign title. It is worth noting that it was bilingual in Chinese and Manchu, but the readings (to a mostly Chinese audience) would take place in the Chinese language rather than Manchu – the rites were, in any case, a Ming holdover.

This was more or less the standard model between 1735 and 1908. The sequence in Puyi's accession is badly mangled in Bertolucci's The Last Emperor, as you can see the sealing of the edict of accession is shown before the kowtow of those in attendance, and the critical step, the ascension to and sitting upon the throne, receives little notice at all.

If you can decipher the Classical Chinese, you can find the version in the statute books here, while an English-language translation of the official description of the Daoguang Emperor's accession in 1820 can be found here. Evelyn Rawski, in her chapter ‘The Creation of an Emperor in Eighteenth-Century China’ in the book Harmony and Counterpoint: Ritual Music in Chinese Context (1996), gives a condensed account of the ritual proceedings of the Qianlong Emperor's accession in 1735, and discusses their symbolism and context. A brief summation of this can also be found in her book The Last Emperors (1998).