Why is Uranus the only planet in our solar system (excepting Earth, of course) that has moons NOT named after Roman or Greek deities?

No one is certain where it started, but it is now tradition, a tradition that started in 1948. The "Literary Moons" are mostly named after Shakespearian characters, with another author thrown in...

https://theworld.org/stories/2017-01-01/why-moons-uranus-are-named-after-characters-shakespeare

Yes, Shakespeare and all that, but WHY are Uranus' moons named differently? The reason goes back to the planet's discoverer and the planet's original name, which was not Uranus.

William Hershel discovered Uranus in 1781 and originally wanted to name it in honor of King George III. Of course everyone outside of the British empire was not happy with the name. It wasn't until 70 years later that general agreement on the name Uranus was decided. In 1852, William Hershel's son decided to name the first two moons his father discovered. His friend William Lassell may have been to one to originally wanted the poetic character names. As more and more moon were discovered the tradition continued since the discoverers often are allowed to name their discoveries and they have chosen to continue with the Shakespeare and Pope names.

So, there's reason to believe that when his contemporary William Lassell stumbled upon two more of Uranus' moons in 1851, John Herschel may have had a hand in theirnaming, too.

In 1899, William Lassell's daughter Jane Lassell told a reporter that the moons' names were given by Sir John Herschel, "to whom my father applied." What that means for sure, we'll never know - she also said she lost their letters in a move - and Herschel never took the credit.

https://theworld.org/stories/2017-01-01/why-moons-uranus-are-named-after-characters-shakespeare