[Urwen]

Sure. Here are my explanations for each:

Tom
To some, I am but fairy tale, - Not much is really known about Tom
The subject of your jest. - see below
To others, few, who know me well,
I took an endless quest. - He had various adventures

With slander, mockery, and slur
You tarnish my good name. - Some hobbits have a distaste for all things outside of the Shire, including Tom
However, one thing you have gotten sure:
Punctuality's not my claim. - He does things at his leisure

Relic old from ancient times - Tom is known as the oldest
I guard against all foes. - he doesn't seem to be afraid of anything
My charge to you revealed designs,
To others - secret words. - this one I really can't fit here.

Treebeard
To some, I am but fairy tale,
The subject of your jest. - Ents are not generally known
To others, few, who know me well,
I took an endless quest. - To find the Entwives

With slander, mockery, and slur
You tarnish my good name. - Some of the Shire-folk did in the first book.
However, one thing you have gotten sure:
Punctuality's not my claim. - He isn't hasty, to be sure.

Relic old from ancient times
I guard against all foes. - He protects the trees
My charge to you revealed designs,
To others - secret words. - Saruman's designs

Gandalf
To some, I am but fairy tale,
The subject of your jest. - To Hobbits, he is primarily known for his tricks and jokes
To others, few, who know me well,
I took an endless quest. - To stop Sauron, which presumably took a long time

With slander, mockery, and slur
You tarnish my good name. - At least in Gondor and Rohan
However, one thing you have gotten sure:
Punctuality's not my claim. - He comes and goes at his leisure

Relic old from ancient times
I guard against all foes. - It's in his job description
My charge to you revealed designs, - His sojourn with Saruman revealed Saruman's true designs
To others - secret words. - The inscription on the Ring and the one on doors to Moria.