How often and over what period is Earth’s rotation averaged to compute UT1? I understand that UT1 (and for that matter UT0, UT2, etc.) are based on averages of actual earth rotation, and serve as a form of mean solar time. However it’s not clear to me when these averages are taken and over what range? Certainly the average is taken over at least a year; but beyond that I can’t tell. Daily over the year? Yearly over the year; over several years?

How often and over what period is Earth’s rotation averaged to compute UT1?
Conceptually, the average is over an average year. Practically, that's not what's done anymore.

Sidereal time is much more readily measured than is solar time. Radio astronomy can measure the position of remote quasars much, much more precisely than they can measure the position of the Sun. The Sun is a big fiery blob with a not quite-well defined surface. Quasars are pinpoints. Very long baseline interferometry makes the observations of those quasars extremely precise.

The equation of the equinoxes converts between apparent and mean sidereal time. A very simple relation converts between mean sidereal and mean solar time. This makes variations in UT1 observable on a rapid basis. Sub-arc second astronomy depends on these rapid updates. IERS Bulletin A is updated daily, with the data for the most recent week or so subject to subsequent small updates.

Update

Universal Time is not based on apparent solar time. Time is now effectively divorced from the the apparent revolution of the Earth with respect to the Sun. It took several thousands of years to make this divorce final. (And the divorce is still not quite final. There's the matter of leap seconds. There's a proposal to eliminate leap seconds that has been shelved/postponed multiple times since 2003. A vote is scheduled for 2015.)

The ancient Egyptians divided daytime into 12 equal parts and nighttime into 12 equal parts. Daytime and nighttime hours were of different lengths, and these lengths varied over the course of a year. Hipparchus (190 – 120 BCE) suggested dividing one day into 24 equal parts, the basis for our modern hour. Ptolemy, using information already garnered by ancient Babylonians, asked "what day"? The length of a solar day (local noon to local noon) varies over the course of a year thanks to what we now call the equation of time.

This was of mostly academic interest until Christopher Huygens' development of the pendulum clock in 1656. Pendulum clocks tick at a rate proportional to the mean solar day rather than the apparent solar day. Those early pendulum clocks weren't particularly accurate. They had to be reset on a regular basis. The equation of time was central in determining the time to which those clocks needed to be set.

By the mid 1800s it was becoming apparent that even a mean solar day was not that good a basis for defining time. The clock kept by the solar system didn't agree with a clock kept by the Earth's mean rotation. A clock based on the solar system was more accurate and more consistent than was a clock based on the Earth's rotation.

By 1900, the divorce between Earth rotation and time was well on its way to being finalized. The basis for time was switched from the Earth's rotation to the clock kept by the solar system, with the second based on the length of a year rather than the length of a day. About 150 years of observations of the length of the mean solar day went into that circa 1900 definition of the second.

In 1967, time was further divorced from the clock kept by the solar system. By that time, atomic clocks had become extremely accurate devices, better than observations of the solar system. The International Atomic Time second was defined to be consistent with that circa 1900 definition of a second, which in turn was defined to be consistent with that ~150 years of observations of the length of an apparent solar day, centered on about 1821.

That 1821-based definition of a second as 1/86,400 of a day is no longer correct. A day is now about 86400.002 seconds long, and it will grow longer with passing millennia. This leads to the need to insert leap seconds to keep UTC in synchronization with the Earth's mean rotation. There is considerable debate with regard to whether even that should be done. There is a serious proposal to abandon leap seconds. This is now scheduled for a vote in the upcoming year (2015).