The original listing of these languages is so much more complicated than this, because Tolkien always changed or created new dialects.
Tolkien's glossopoeia has two temporal dimensions: the internal (fictional) timeline of events described in The Silmarillion and other writings, and the external timeline of Tolkien's own life during which he continually revised and refined his languages and their fictional history. He constantly devised and changed the languages and their chronological order changes in real time.
- 1910–1930: Primitive Quendian the proto-language, Common Eldarin, Quenya and Goldogrin.
- 1935–1955: Goldogrin had significantly changed and was now Noldorin, joined by Telerin, Ilkorin, Doriathrin and the Avarin.
- The late and mature stage dispensed with Ilkorin and Doriathrin. Noldorin matured into Sindarin.
Tolkien had worked out much of the etymological background of his Elvish languages during the 1930s (collected in the form of The Etymologies). In 1937, he wrote the Lhammas, a linguistic treatise addressing the relationship of not just the Elvish languages, but of all languages spoken in Middle-earth during the First Age.
- 1940s, while working on Lord of the Rings, Tolkien invested great effort into detailing the linguistics of Middle-earth.
- Mannish languages and Dwarvish was created between 1940 and 1945.
- However, in 1945, Tolkien devised the language known as Adûnaic. Its development began with The Notion Club Papers (written in 1945). It is there that the most extensive sample of the language is found.
- Tolkien remained undecided whether the language of the Men of Númenor should be derived from the original Mannish language (as in Adûnaic), or if it should be derived from "the Elvish Noldorin" (i.e. Sindarin) instead.
But first creation of these languages (which makes the chronological order in some sense) looks like this:
- Primitive Quendian
- Common Eldarin
- Avarin (6 languages)
- Goldorin >> Noldorin
- (Common)Telerin (has dialects)
- Noldorin >> Sindarin
- Northern Mannish
- Taliska (2 dialects)
- Adûnaic (there is a tricky part)
Most of the "first 15" include sub-languages and dialects as well. So if you include them all, Adûnaic cannot be listed as the 15th language because it is a sub-language-like language that was derived from the Hadorian tongue. It is under Taliska language branch but I listed it as the 15th. Because even if it's a sub-language or a dialect, if it's talked by masses it is considered as a language.
The language of the House of Beor eventually was almost extinct, both because of the adoption of Sindarin and because the House was mostly annihilated after the Dagor Bragollach. On the other hand, the House of Hador did not wholly abandon their language, it survived the War of Wrath and was still spoken when the Edain migrated to Elenna. Thus it later became Adûnaic.
Adûnaic, the language of Númenor, developed in 1946 while he was finishing up Lord of the Rings, was said to be his fifteenth invented language.
You can see the language tree here.
Creation of languages didn't happen in chronological order of the stories and books. Tolkien simply did that when he was in need of a new language.
Adûnaic was generally considered to be a language of less prestige than the Elven tongues. Most locations of Númenor, and most of the lords and ladies of the Dúnedain, had also Quenya or Sindarin names beside their native ones. Even most commoners knew Sindarin to some degree.
Adûnaic was derived from the Hadorian tongue, related to the Bëorian—collectively called Taliska. It was more distantly related to the languages of the Middle Men of the east, such as the Northmen.
During the First Age these languages were much influenced by Khuzdul, Avarin, but also by the languages of the Eldar, as the Elf-friends spoke Sindarin.
Taliska was not related to the Haladin tongue at all, therefore when the Númenóreans returned to Middle-earth in the Second Age, they did not recognise the peoples of Enedwaith and Minhiriath as their distant kin, who spoke unrelated languages.
You can visit here for a language analysis of the 15th language.
- Part from Sauron Defeated by C. Tolkien:
When The Lord of the Rings had still a long way to go - during the halt that lasted through 1945 and extended into 1946, The Return of the King being then scarcely begun - my father had embarked on a work of a very different nature: The Notion Club Papers; and from this had emerged a new language, Adunaic, and a new and remarkable version of the Númenórean legend, The Drowning of Anadûnê, the development of which was closely entwined with that of The Notion Club Papers.
Sauron Defeated, HoME 9, "Foreward"
How is all this to be equated with his statement in the letter to Stanley Unwin in July 1946 that 'three parts' of the work were written in a fortnight at the end of 1945? Obviously it cannot be, not even on the supposition that when he said 'a fortnight' he greatly underestimated the time. Though not demonstrable, an extremely probable explanation, as it seems to me, is that at the end of that fortnight he stopped work in the middle of writing the manuscript E, at the point where The Notion Club Papers end, and at which time Adunaic had not yet arisen. Very probably Part One was at the stage of the manuscript B.4 On this view, the further development of what had then been achieved of Part One, and more especially of Part Two (closely associated with that of the Adunaic language and the writing of The Drowning of Anadûnê), belongs to the following year, the earlier part of 1946.
Sauron Defeated, HoME 9, Introduction to The Notion Club Papers
- You can find most of the above in J.R.R. Tolkien, Christopher Tolkien (ed.), The War of the Jewels, Part Four "Quendi and Eldar": Appendix D: *Kwen, Quenya, and the Elvish (especially Ñoldorin) words for 'Language': Note on the 'Language of the Valar', p.397.
Addition: Since the OP insists on the existence of a chronological list of invented languages, made by Christopher Tolkien, I double-checked my copy of the volume. Couldn't find one (if you find any, please share it with a page number or some sort of referable source). But the book examines and tells the story of a newly discovered language called Adûnaic.
How was it counted as the 15th invented language if there isn't a list in Christopher Tolkien's hand? I believe Christopher Tolkien simply counted the early 14 and added this one as the 15th.