For those have read The Adventures of Tom

[The Squatter of Amon Rûdh]

Time for a resurrection. It's been a long while since this question was asked, but perhaps some other people have felt the same confusion.

It appears that the poem in question is No. 7, The Stone Troll.

As it happens, we can't take "Uncle Tim" as a mere respectful term of endearment, because in the fourth stanza we have

Said Tom: 'I don't see why the likes o' thee
Without axin' leave should go makin' free
With the shank or the shin o' my father's kin...'

The explanation is quite simple. Not all of the poems in Adventures are about Tom Bombadil, which makes it quite a misleading title. Tolkien, in the character of an editor, explains their various origins in the preface to the collection.

The Red Book contains a large number of verses. A few are included in the narrative of the Downfall of the Lord of the Rings, or in the attached stories and chronicles; many more are found on loose leaves, while some are written carelessly in margins and blank spaces...

The present selection is taken from the older pieces, mainly concerned with legends and jests of the Shire at the end of the Third Age...
In the Red Book it is said that No. 5 was made by Bilbo, and No. 7 by Sam Gamgee.

This poem was adapted from a much earlier verse called The Root of the Boot, which Tolkien wrote while teaching at Leeds. The protagonist's name is unchanged from this first version, which was composed a little earlier than Tom Bombadil's first appearance (in The Oxford Magazine, Vol. 52 No. 13, 15th February 1934).

Of course, just the few lines quoted above will identify this poem to most people as the one Sam recites in The Flight to the Ford. The name of the protagonist is coincidental, having been a very common personal name for centuries.