[1] Siddur Admur; M”E 582/1 and 22; Piskeiy Teshuvos 582/9; not found in earlier sources.

The reason: See Halacha 5D and footnotes there. To note however that this change of dialect in Kaddish is not found in previous sources and thus is disputed against, as explained next.

Other opinions: Some write that according to the Kabalists one is not to change the dialect in Kaddish and is rather to always say Oseh Shalom as is said during the year. The change is only made by the end of Shemoneh Esrei. [Kaf Hachaim 56/38; Alef Hamagen 582/37]

Custom of Sefaradim: The Sefaradim are accustomed to only recite Oseh Hashalom in Kaddish Tiskabel that follows Shemoneh Esrei. They do not recite it by other Kaddeishim. [Moed Lekol Chaiy 13/69; Piskeiy Teshuvos 582/9; See Yalkut Yosef Moadim p. 26 ]

[2] Siddur Admur [no difference of Nussach is mentioned]; Gloss of Tzemach Tzedek [printed in Shaar Hakolel 4/9]; Piskeiy Hassidur 171; Sefer Haminhagim p. 118 [English]; Kaf Hachaim 582/39 based on Arizal [regarding all Aseres Yimei Teshuvah, however including Neilah] and so is the custom of the Sefaradim [Piskeiy Hassidur ibid]; Shaar Hakolel 4/9 based on Seder Hayom that explicitly says to not say it and Mishnes Chassidim and Siddur Yaakov Kopel that omits it.

The reason: The dialect of Kaddish contains exactly 28 words according to Kabala. Based on this one is not to change the dialect even during Aseres Yimei Teshuvah. [Kaf Hachaim ibid] However the M”E 582/1 writes that when Lieila is repeated the word Min Kol Birchasa is omitted and read as Mikol Birchasa, and hence there remain only 28 words.

Ruling of Admur in Shulchan Aruch: In Shulchan Aruch of Admur 582/16 he rules that one adds “Lieila Lieila” throughout all the Kaddeishim recited during Aseres Yimei Teshuvah.

Other Poskim: The following Poskim write one is to recite Leila Ulieila throughout Aseres Yimei Teshuvah: Levush 582/8; M”A 584/4; Admur ibid; M”B 582/16; M”E 581/1; Siddur Yaavetz; Kitzur SHU”A 129/1; Darkei Chaim Veshalom 706; Luach Eretz Yisrael. All the above Poskim write that one is to repeat “Lieila Lieila”. However in some Sefarim it says “Lieila Ulieila”. [Likkutei Mahrich; Piskeiy Teshuvos 582/9] The reason for repeating the word Leila is because during Aseres Yimei Teshuvah Hashem sits in judgment and His name is hence glorified before the masses much more than the regular year. Hence they emphasize that He is very exalted during this time. [Machatzis Hashekel 582/4 in name of Levush ibid] Some Poskim rule it is only to be repeated on Rosh Hashanah and Yom Kippur. [Elya Raba 582]

[3] Sefer Haminhagim p. 125 [English]; Hayom Yom 10^th Tishrei; Kuntrus Hashulchan p. 63 based on Mamar of Rebbe Maharash [Toras Shmuel 1874 p. 284]; See Otzer Minhagei Chabad 262

 Other opinions: The Shaar Hakolel 4/9 states that it is not to be repeated at all even during Neilah and that doing so has no source.