The French, the British, the Germans, the Scandinavians and the Baltic Finns call Christmas with the following word that remains from pagan times: Yule (Eng.), Jul, Julen (Sw., Dan., Norw.), Jól (Icelandic), Joulu (Finnish, Izhorian), Jõulu(d) (Estonian, Votic) - allegedly considered to be a word 'of unknown origin': https://www.etymonline.com/word/yule .
Noteworthy, however, is that the followers of the Zoroastrian (Persian, Iranian) tradition use the same name when celebrating the winter Solstice: they call the longest and darkest night of the year as Shab-e Yalda, or Shab-e Chelleh (Çillə). [Spoiler (click to open)]In Zoroastrian tradition it is considered to be a particularly inauspicious night when the evil forces of Ahriman are imagined to be at their peak. One is advised to stay awake most of the night, to avoid any misfortune. People gather in safe groups of friends and relatives, and share their last remaining fruits of the past summer. The following day (the first day of Dae month) is a holiday. The word Yalda supposedly means 'the Birth' or 'to give birth'.
It is further comparable to: [yuladu] يولد (Arab.), [yalad] יָלַד (Hebrew) - to give birth; [hуlad] הוּלַד (Hebrew) - was born; [yalud] יָלוּד (Hebrew) - newborn[Spoiler (click to open)]; [eled; yaldo] יַלדו ; יֶלֶד (Hebrew) - a boy, a child - i.e. the infant Sun, which is 'born' each year at winter solstice). Source: https://ich-neu-mon.livejournal.com/68150.html.
It is obvious that the Birth of the new Sun is exactly what the name of the holiday, the Yule, reflects.
The related words in other languages include:
Лада, Леда / Lada, Leda (Slavic) - the God's Mother, Our Lady (Eng.); [ləda] לֵידָה (Hebrew) - childbirth, parturition.
плод [plod] (Rus.) - a fruit, an offspring. ялта, ялат, елат [yalta, yalat, elat] (Ingush) - grain: child (Eng.).
кылды(ны) [kõldõ(nõ)] (Udmurt) - to be born, to appear; кылдыты(ны) [kõldõtõ(nõ)] (Udmurt) - to give birth to, to create; Kõldõs(in) (Udmurt, Komi), калтась(эква) [kaltas'(ekva)] (Khanty) - the Creator God(Goddess), the Forefather (the Foremother) of Nature.
[gal] (Sumer.) - to be, exist; ol(mak) (Turk.), ol(maq) (Azeri.), bo'l(moq) (Uzbek), болуы [boluõ] (Kazakh) - to be, to live, to be born; ole(da) (Est.), olla (Est., Fin., Izhor., Karel.), õlla (Votic), vȱlda (Livonian), olda (Karel.), ouda (Veps.), uľems (Erzia), uľǝms (Moksha), ulaš (Mari), ol-, ɔ̄l- (Khanty, Mansi) - to be, to exist, to possess; ela(da) (Est.), elää (Fin., Votic), ellää (Izhorian), eliä (Karel.), eľädä (Chudi, Veps.), eallit (Saami), ilaš (Mari), uli̮ni̮ (Udmurt), ovni̮ (Komi), él(ni) (Hung.), ńile (Nganasan), ela- (Sekulp.), je’llõ (Livonian), jiľe (Nenets.) - to live, to be; ďili, ďž́ili (Kamassian), elävä (Fin.) - alive, living; jiləp (Khanty) - new, young; йель [yel'] (Saami), elu (Est.), olan (Komi) - life, essence; yıl (Turk.), il, ildir (Azeri), yil (Uzbek), ел [yel] (Tatar), жыл [zhyl] (Kaz., Kyrgyz), сол [sol] (Tadzhyk) - a year (the semantics of a life cycle); elo (Tagalog in Philippines) - life.
There is also an apparent connection of all of the above words with the Sun and Sunlight:
Hλιος, Helios (Gr.) - the Sun; the Sun God; Gela, Gal'Erd - the Sun God in Vainakh (Ingushian) pagan tradition in the Caucasus; hele, heleda (Est.), heleä (Fin.), õ’ldzi (Livonian), eliä (Votic), helliiä (Izhor.), heľei (Karel.), heleäd (Chudi), heled (Veps.) - 1) light, bright; 2) lively, sonorous; [elel] (Maya) - to burn; al [ал] (Turk., Azeri, Tatar, etc.) - scarlet, bright red, crimson; alev (Turk., Crimean Tatar), alov (Azeri), ялын [yalyn] (Kumyk) - a flame, fire; алау [alau] (Kaz.) - a bonfire; аьл [a'l] (arch. Ingush.) - the Sun; ал [al] (Ingush., Chechen) - a fire; ала [ala] (Ingush.) - a flame; аълан [a'lan] (Ingush.) - to burn; ælan (arch. Eng.) - to burn; eld (arch. Sw.), ild (arch. Norw., arch. Dan.), eldur (Icelandic, Faroese), aile (Manx) - a fire, flame, bonfire; ilde (Norw.) - to heat, to keep fire; ἀλέα [alea] (arch. Gr.) - the heat; ἀλεεινός (arch. Gr.) - warm, heated (by the Sun); алый [aly] (Rus.) - scarlet, bright red, crimson.
sole (Ital.), sol (Latin, Sp.), soleil (Fr.), sol, solen (Sw., Dan., Norw.), sól, sólin (Icelandic), солнце [solnce] (Rus.) - the Sun; soləm, suləm (Chuvash), жалын [zhalyn] (Kazakh) - a flame, fire; [su.lum, su.lim] (Sumerian) - a radiation, splendor.
КОЛЯДА, KOLYADA, CALENDAE
The Slavic old name for Christmas festivities (celebrated over 12 days, from Christmas to Epiphany) was Коляда, Колѧда, Kolyada, Kolianda, for which the Slavists (Max Vasmer) suggested a 'borrowing' from Latin Calendae, the Greco-Roman holiday celebrated on 5 January, i.e. at Epiphany, 12 days after Dies Natalis Solis Invicti - and from which we have inherited the word calendar.
It is obvious that the Slavic Kolѧda and Roman Calendae are one and the same name for the same winter solstice celebrations. Yet, it is naive to associate the etymology with mere 'shouting' at festivities, or to a public 'announcement' of the beginning of a new month / new year. According to a number of independent researchers, the custom has a deep pagan origin, and the etymology is to be re-studied: https://ru.wikipedia.org/wiki/Коляда .
Коляда, Колѧда, Kolyada, Kolianda / Calendae may well be connected to all or one of the following:
1) The Germanic Yule, the Estonian Jõulu(d), the Persian Yalda - Christmas - reviewed above.
2) The Slavic коло, kolo - a wheel, a circle, the Sun circle, as well as sol, solis (Latin) - the Sun - analyzed above.
3) The Russian колода, koloda - a log - a synonym of the Southern Slavic бадняк, badnyak - the Christmas block, burnt over 12 nights after Christmas.
4) The Russian холод, kholod, Eng. cold - and a huge semantic field around these words, reviewed below. Of notice is the Bulgarian name of Santa Claus - Дядо Коледа, Dyado Koleda - effectively, Father Frost.