Is there an established notation, either modern or historical, for any unit of measure which is then further subdivided into 360 degrees or parts?

In general? I won't go so far as to rule it out completely, but I'll say with moderate confidence there probably isn't.

It's probably worth pointing out that while the Ancient Babylonians are credited with originating the 360 degree circle, this actually a half-truth that glosses over a point relevant here. They didn't actually have any concept of angle or arclength as numerical magnitudes. They just had length. But they could take some unit length, use it to describe a unit circle of radius 1, and simply construct an inscribed regular hexagon of perimeter 6 inside the circle, and see that the circumference of the circle must be slightly greater than 6. Since their numeral system was sexagesimal, they grouping and partitioning of units was by 60s, so if they wanted to refine the resolution of length measurement by an order of magnitude they would use 1unit = 60parts and get 6units = 6*60parts = 360parts.

Now in answer to your question, we might write 60º = 1', 60'= 1'' and so on. This superscript notation comes from the Romans. You'll note that the primes are in fact the Roman numerals for one and two, and the degree symbol is indeed a zero glyph which is functionally a decimal point.

You could call it a "decasecond" since $10'' \cdot 360 = 3600'' = 60' =1^\circ$ or $1$ hour.

Also it is almost the number of days in a year.

There does not appear to be an established notation for the unit circle, but there are pretty close things.

The notation of $^\circ, ', ''$ derives from using roman numbers as column markers, as one might write 5h 3t 6 for five hundred, thirty, six. It is not specific to base 60, since the same scheme is used of feet, inches and lines (and downwards to points), on a duodecimal scale, for the french grade, using centisimal scale, and by the earliest decimalists, to denote dimes, cents and mills.

The measure at $^\circ$ is a unit. When one writes $1^\circ 30'$, one is writing 1.5 units. However, there are some interesting arguments that the circle, and not some fraction ought be the unit of measure.

When one measures angle, it is usually reckoned as a fraction of surface, measured in measures of the radius. The Sumerian systems suppose that $\pi=3$, and has $2\pi \cdot 60$ degrees for circles in the sky, and $\pi \cdot 60$ ells of $24$ digits for real circles (like things you can walk around). This is in Sir Thomas Heath's 'history of greek mathematics'.

In the higher dimensions, one might want the same angle preserved when a cartesian product of full space is applied: that is, the angle between the planes is the same as the solid angle. This happens when all-space is taken as $1^\circ$, and the primes, seconds, etc refer to fractions of it. So the angle where two square faces of a hexagonal prism meet is the same as the hexagon corner-angle, ie 0;40 = 1/3.

The measures are made in base 120, which greatly simplifies hand calculation, and because the first division gives the 12 hours of the clock, simplifies that too. Base 120 is historically attested in England, see eg 120 on the wikipedia for references.

Aslo, because a complete circle is shown with a circle-rune $^\circ$, it some how makes some sense.