Who declares the end of the end of the Era in the Elder Scrolls series

Generally speaking, whomever is in charge

Dawn Era → Merethic Era

In the Dawn Era, the gods were actively creating Mundus and time worked non-linearly. Therefore, it's impossible to measure it. The Merethic Era begins when time moves linearly, and it's marked by the prevalence of Elves.

With Magic (in the Mythic Sense) gone, the Cosmos stabilized. Elven history, finally linear, began (ME2500).

_{—Before the Ages of Man, first appearing in Oblivion}

Merethic Era → First Era

In the Nordic language, "Merethic" means "Era of Elves", and marks the time when the Aldmer (ancestors to modern elves) ruled. The transition between eras happens at the point when the Nords consider to be the "beginning of time".

The Merethic Era was figured by early Nord scholars as a series of years numbered in reverse order backward from the their [sic] 'beginning of time' -- the founding of the Camoran Dynasty, recorded as Year Zero of the First Era.

_{—Before the Ages of Man, first appearing in Oblivion}

First → Second Era

After the last Emperor of the Reman Dynasty is assassinated, Akaviri Potentate Versidue-Shaie takes the throne and declares the start of a new era.

As my first act, I [Potentate Versidue-Shaie] declare that in commemoration of this historical moment, beginning on the first of Morning Star, we will enter year one of the Second Era as time will be reckoned. Thus, we mourn the loss of our Imperial family, and look forward to the future.”

_{—2920, The Last Year of the First Era, first appearing in Morrowind}

Second → Third Era

With the help of the Numidium, Tiber Septim unites all of Tamriel into one Empire. He declares the start of a new era.

The year was 2E 896. The following year, the Emperor [Tiber Septim] declared the beginning of a new Era-thus began the Third Era, Year Aught.

_{—A Brief History of the Empire, first appearing in Morrowind}

Third → Fourth Era

UESP.net says that the transition happened with the Oblivion Crisis (i.e. The Elder Scrolls IV: Oblivion).

The Septim bloodline ends and the Amulet of Kings is destroyed. The Third Era ends, marking the beginning of the Fourth.

Some of the other wiki pages repeat this and cite the "Events of Oblivion". I'm not sure if there is in-game character dialogue or something regarding this; I'm not able to find a direct quote about who decided this transition.

It's possible that by this point, the end of a dynasty had now traditionally marked the end of an era. More information is needed to clarify this.