In terms of other nations observing a sabbath, this is divided into two questions
Which nation observed a seven day week. Once you have the week, you will mark the seventh day as a special day with prescriptions and prohibitons on that day. Indeed, that's what it means to observe a week.
Of those required observances, how similar were they to the Mosaic regulations?
Seven Day Week
The seven day week goes at least far back as ancient mesopotamia, with the first recorded example of a general seven day week proclamation issued by Sargon of Akkad. But even that proclamation was a reform aimed at standardization, so we must assume there were weekly observances that were not harmomized when Sargon expanded his empire. Thus the week is likely far older than Sargon's reign and there is evidence it predates it:
The earliest evidence of an astrological significance of a seven-day
period is connected to Gudea, the priest-king of Lagash in Sumer
during the Gutian dynasty, who built a seven-room temple, which he
dedicated with a seven-day festival. In the flood story of the
Assyro-Babylonian Epic of Gilgamesh, the storm lasts for seven days,
the dove is sent out after seven days, and the Noah-like character of
Utnapishtim leaves the ark seven days after it reaches the firm
ground. [c] Counting from the new moon, the Babylonians celebrated the
7th, 14th, 21st and 28th as "holy-days", also called "evil days"
(meaning "unsuitable" for prohibited activities). On these days,
officials were prohibited from various activities and common men were
forbidden to "make a wish", and at least the 28th was known as a
In Mesopotamia, however, we must understand the week to be a subdivision of the month rather than as an independent unit. A quarter of the lunar mont (e.g. when the moon goes from dark to half full) is 7.375 days, and indeed in Babylonia they would extend the last week with extra days as needed to keep the weekly calendar synced up with the lunar calendar so that one lunar month was four weeks.
Another conjectured source of the week was from astronomy, with the days given in honor of the sun, moon, and five planets. Indeed many days of the week are named after these seven heavenly bodies.
These two conjectured sources may not be in conflict - perhaps the units were chosen as a subdivsion of the month and merely named after the heavenly bodies. But this is speculation.
But most belive that a seven day week originated in ancient Sumeria, then spread out to Assyria and then the other cannanite tribes and Israel. It would certainly make sense that Abraham, growing up in Babylonia, would be familiar with the week and then pass it on to his descendants. Indeed the word for "week", and "rest" are all closely related in ancient semitic languages to the common word for seven or "sheba" and all semitic cultures viewed this as a holy number, and would thus view the seventh day as a holy day.
the seven days reckoned as the length for eating unleavened bread in
all likelihood represents the heritage of Canaanite culture, which in
this instance, however, functions merely as the mediator for a notion
found everywhere in the OT, since all the cultures in the OT environs
understand the number seven as a sacred number and as an expression of
Haag, E. (2004). שַׁבָּת. G. J. Botterweck, H. Ringgren, & H.-J. Fabry (Eds.), D. W. Stott (Trans.), Theological Dictionary of the Old Testament (Revised Edition, Vol. 14, p. 390). Grand Rapids, MI; Cambridge, U.K.: William B. Eerdmans Publishing Company.
The Hebrew noun šabbāṯ occurs as a loanword in Aramaic (st. abs.
šabbāʾ, st. determined or emph. šabbeṯāʾ pl. šabbayyāʾ), Syriac (st.
abs. šabbā, st. emph. šabbeṯā, pl. šabbîn), Arabic (sabt), and
Ethiopic (sg. sanbat, pl. sanābet and sanbatāt).
Haag, E. (2004). שַׁבָּת. G. J. Botterweck, H. Ringgren, & H.-J. Fabry (Eds.), D. W. Stott (Trans.), Theological Dictionary of the Old Testament (Revised Edition, Vol. 14, p. 389). Grand Rapids, MI; Cambridge, U.K.: William B. Eerdmans Publishing Company.
In Assyria also, the common units of time were the day, the week (hamuštum), half-month and month.
The commonly accepted value of the hamuštum, a week, was computed by
K. R. Veenhof in 1996. The calculation of its length was made possible
from a few texts that give at the same time the amount of the debt,
the interest rate, the amount of the interest and the number of
hamuštum for which the interest is due after the settlement date has
expired. The result could be compared to the unique hamuštum almanach,
Kt g/k 118, which seems to give a complete list of 50 to 52 hamuštum
corresponding to a single year led to a hamuštum period of a seven-day
week. Each hamuštum of this list is named for two merchants.
Differences between Hebrew and Babylonian week
In Babylon, the final week of the month was extended as needed to match the lunar month, so the holy days could be 7, 14, 21, 29. It was these types of strategies that required standardization in Sargon of Akkad's reforms.
In the Mosaic law, the week was always seven days and allowed to drift away from the lunar month. So the holy days would be: 7, 14, 21, 28 and they would have separate lunar sabbaths that would be inserted (say on the 29th or 30th day) that didn't fall on the seventh day. Indeed, holy days were considered "sabbaths" whenever they fell, and there is evidence that the original feast days were celebrated according to lunar reckoning, so here too there is some conflict between the number 7 and the lunar cycle, but in the Mosaic Law the conflict was resolved by adding in more sabbaths rather than removing some as the Babylonians did.
The Sabbath day and rest
It was common to prohibit work on the holy day, for obvious reasons -- e.g. it was a holy day. But for the Babylonians work was prohibited because the day was considered unlucky, and thus bad for work. On the other hand, the Mosaic Law has a more positive description, emphasizing rest for the sake of resting.
All of the Mosaic Law should be viewed as reforming existing practices that were widespread rather than inventing brand new practices out of whole cloth. For example circumcision is practiced all around the world and dates to the stone age (hence the use of flint knives). All of the animal sacrifices, for example, were already in widepsread practice, especially in Canaanite religion. The use of cherubim in the temple, the concept of a most holy place, holy garments for priests, sacred numbers, the notion of an "ark", the idea of feast days to commemorate harvest, the new year, etc. These were all widespread, as was the week and the concept of a sabbath at the end of the week. The same held for purity laws, divorce laws, etc. We should view the revelation of Moses as reforming existing practices that were well known to everyone involved, which is why you don't need explanations on what circumcision is or how to do it, or you don't have a lot of explanations about animal sacrifices or how to calculate new moons -- people throughout the region were already well-versed in all these things. What Moses did was regulate and refine these practices, putting them into a coherent Law that directly tied into Israel's position as the bride of Yahweh.
Indeed this is consistent with the Biblical view that the Sabbath was instituted at creation, and thus known to Adam, and passed on through his descendents, together with the traditional notion that Revelation has been declared to the whole world (Psalm 19), but for the revelation declared to the whole world, it is mediated through a veil of visions and dreams and voiceless speech, whereas the revelation to Moses was clear and direct:
And Yahweh went down in a column of cloud and stood at the doorway of
the tent, and he called Aaron and Miriam, and the two of them went,
and he said, “Please hear my words: If there is a prophet among you,
I, Yahweh, will make myself known to him in a vision. I will speak to
him in a dream. Not so with my servant Moses; in all my house he is
faithful. I will speak to him mouth to mouth, in clearness, not in
riddles; and he will look at the form of Yahweh. Why were you not
afraid to speak against my servant, against Moses?”