The comments of A. M. Bittlingmayer convinced me my quick answer based on Wikipedia was insufficient. I then ran a small script through the CLDR (actually a python script using the Babel package) to output a given dates in all included languages and sort out the “Big endian” ones. It confirms my old answer, and completes it.
Of course, it is just as good as the CLDR data, and more knowledge of the languages would be needed to validate this. The list below goes beyond Europe, because 1. why not? 2. The borders of Europe are difficult to define, and “big endian” languages in Russia (Uzbek, Chechen). It gives the
'long' date format of CLDR for “”, which is intended to reflect the way it is spoken. The corresponding output for English is “February 19, 2018” (“19 February 2018” for British English).
This list also misses “partially big endian languages”, like Azerbaidjani where January 13, 2012 is 13 yanvar 2012, but January 13 – 14, 2012 is indeed 2012 yanvar 13–14.
- Hungarian: 2018. február 19. (checked on Wikipedia, see old answer below)
- Lithuanian: 2018 m. vasario 19 d. (checked on Wikipedia, see old answer below)
- Chechen: 2018 февраль 19
- Inari Sami: 2018 M02 19 (the form seems a bit suspect, but 1. Chinese use the same kind of form, since the month do not have Chinese names, only numbers; 2. Northern Sami seems to be big-endian too, but with month names.)
- Northern Sami: 2018 guovvamánnu 19
- Breton: 2018 Cʼhwevrer 19
- Basque: 2018(e)ko otsailak 19
- Church Slavic: 2018 феврꙋа́рїа 19
Central Asian Languages (including Asian Russia)
- Uzbek: 2018 Феврал 19
- Kyrgyz: 2018 февраль 19
- Sakha: 2018, Олунньу 19
East Asian Languages
- Chinese, Cantonese, and Japanese : 2018年2月19日
- Korean : 2018년 2월 19일
- Sichuan Yi: 2018 ꑍꆪ 19
- Dzongkha: སྤྱི་ལོ་2018 ཟླ་གཉིས་པ་ ཚེས་ 19
- Tibetan: སྤྱི་ལོ་2018 ཟླ་བ་གཉིས་པའི་ཚེས་19
- Mongolian: 2018 оны 02 сарын 19
South Asian Languages
- Sinhala: 2018 පෙබරවාරි 19
- Malayalam: 2018, ഫെബ്രുവരി 19
- Nepali: 2018 फेब्रुअरी 19
- Pashto: د 2018 د فبروري 19
- Quechua: 2018 Hatun puquy 19
- Ngomba: 2018 Pɛsaŋ Pɛ́pá 19
- Akan: 2018 Kwakwar-Ɔgyefuo 19
- Metaʼ: 2018 imeg àbùbì 19
- Kinyarwanda: 2018 Gashyantare 19
Artificial Auxiliary Languages
- Esperanto: 2018-februaro-19 (but a quick glance on Wikipedia seems to infirm this)
- Volapük: 2018 febul 19
Hungarian in an example of such a “big endian” European language. See this wikipedia page for example. This wikipedia page in Hungarian uses “1956. október 23-án” for October 23rd, 1956.
Lithuanian also uses a similar format, and this wikipedia page in Lithuanian uses “1990 m. kovo 11 d.” for MArch 11, 1990.
Beyond Europe, this convention is fairly common in East Asia, where it is used in Chinese and Japanese.
According to this map on wikipedia, these languages corresponds to the only European countries where the YMD convention is used exclusively.
Since I know Germans only use DMY colloquially, while this map says both YMD and DMY are used, I guess that DMY there is only “administrative”, and the case might be similar for the other similarly coloured countries.