What it names now
Unvarying describes a thing that stays the same. A rhythm can be unvarying, a daily routine, a climate, a level of attention. Current dictionaries treat the adjective as meaning not changing and not showing variation. Those two readings sit close: one is about time, nothing shifting as hours pass; the other is about sameness of kind, nothing in the set different from the rest. English already has unchanging, constant, and uniform for neighboring jobs. This word is the member of that group built on the verb vary.
The origin of the word unvarying is not a tale of one foreign adjective brought in ready-made. It is a native negative prefixed to a borrowed verb, and the parts arrived centuries apart.
An English compound on a French and Latin verb
The adjective is un- plus the present participle of vary. That is the whole formation. It was put together in English, not taken from Old French as a finished word. First printed evidence falls in the late seventeenth century. One standard etymological dictionary places it in the 1680s; a current American dictionary dates first known use to 1690. The disagreement is a year or so of surviving print, not a clash of stories. First-attestation dates record the earliest citation an editor has found, not the moment a speaker first said the word.
The verb was much older in English. Late fourteenth-century writers used Middle English varien, both to change something and to undergo change. It came from Old French variier, also found in Anglo-French as varier, and, in a parallel channel, from Latin variare. By about 1400 the verb could also mean to differ from a pattern, and, in a sense now obsolete, to disagree.
Negatives of the same stem were already in circulation. Unvariable is attested from about 1400; unvaried from the 1560s. Unvarying is the later, participle-based form. The adverb unvaryingly follows it.
Prefix from the north, stem from the south
The prefix is Germanic. Old English un- is the ordinary negative, from Proto-Germanic un-, from the Indo-European particle that also gives Latin in-, Greek a- and an-, and Sanskrit a- and an-. English kept both un- and in- and sometimes attached them to related stems. Invariable, with the Latin prefix, was already available when this participle adjective appeared. The pair is not a perfect match. Invariable often marks something that cannot be changed. Unvarying more often marks something that is not, in fact, changing: a state, not a prohibition.
The stem is Latin. Vary is from variare, a verb built on the adjective varius. In Latin, varius meant varied and different, and also spotted, mottled, or mixed in color. Writers used it of particolored things, of objects made of unlike parts, and of people who were versatile or fickle. The color sense is early, not a later flourish. It survives in English vair, the heraldic term for a pattern based on squirrel fur, from Old French vair from the same Latin adjective. The everyday family is larger: various, variety, variation, variable, variance.
Romance cognates of the verb are straightforward: French varier, Spanish and Portuguese variar, Italian variare. German variieren is a later learned borrowing of the Latin verb, not an inherited Germanic sibling. English has no native Germanic verb underneath vary; the stem itself is a medieval loan.
Where the trail stops
Latin is as far as a careful account can go. The further origin of varius is unknown. Michiel de Vaan, in his etymological dictionary of Latin, reviews proposed links, including a speculative Italic root shared with vafer, sly or clever, and a derivation from an Indo-European root meaning to leave or let. He finds those proposals unconvincing. A connection with Latin varus, bent outward, has also been floated and is not established. For the English adjective, the honest root is an adjective of difference whose prehistory remains open.
A stable sense, a late date
From its first citations the adjective already means not altering, uniform, unchanging. It did not take on the old sense of vary as disagree. The meaning has been stable; the interesting history is the delay.
Vary is a fourteenth-century loan. Unvariable appears around 1400. Unvaried waits until the mid-sixteenth century. The participle adjective is later still: late 1600s, only a few decades before 1700. The prefix un- had been productive since Old English. English could have formed the word at any time after vary was naturalized. The printed record says it did not, or at least did not often enough to leave an earlier trace. What looks like an everyday not plus varying is, as a documented adjective, a late seventeenth-century formation. That lag, not a shift of sense, is the fact worth keeping.