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Short answer
As a term used to mean a school for preparing young women to enter fashionable society (or something similar), 'finishing school' seems to have already been in use in the late 1790s. There are also several examples of 'finishing school' used in the 1820s, despite several sources stating that these schools (and use of the term 'finishing school') started later (see here for example).
Details
According to Merriam-Webster, the first known use was in 1832. However, looking at the trends of usage in the Collins Dictionary, the first usage was in 1804. Unfortunately, neither dictionary cites where the term was used and the Collins date does not necessarily refer to an institution specifically for young women. kimchi lover found the term used in a 1759 book with reference to clinical physicians (see comment below).
The Irish novelist Lady Morgan uses the term in her memoirs. Although published in 1862, Lady Morgan states specifically the term used at the time she attended (late 1790s) was 'finishing school'. She writes that she was placed
...in the fashionable "finishing school," as it was then called, of Mrs. Anderson....within a few doors of Drogheda House...[in a]...fussy part of Dublin.
From the 1820s, there are several examples. One is for 1828 on the British History Online site. This refers to a finishing school at Acton House in West Acton, London:
Frances Beechey ran a ladies' day and boarding school in 1826 and 1832, described as a boarding and finishing school in 1828 when it was at Acton House, West Acton.
A footnote provides the following source information:
Pigot, Lond. Dir. (1826-7); Com. Dir. (1832-4); Boarding Sch. and Lond. Masters' Dir. (1828), 9.
Also, njuffa (in two comments below) has found two slightly earlier uses of 'finishing school': Joseph Aston, 'A Picture of Manchester' (1826) and Th. Hamilton, 'The Youth and Manhood of Cyril Thornton, vol. 3' (1827).
Other source: