No need to vote on this community-wiki partial possible solution, here only in case it is on the right path and someone else can figure out more of it.

Suppose that each stage of the poem...

...describes a letter in a phrase.

In a land of no name you start your journey with hands stuck at 3,

B looks somewhat like a stick-figure person grasping a 3.

there is a tree with two stumps,

A looks like a ∧-shaped tree standing on two stumpy legs.

a half moon you will see.

D looks very much like a half moon.

So far, so...

B A D,...

...which seems like the start of something to fear. From here, though, things look more confusing than scary.

A fork in the way two roads there must be.

Y (or T) ?

This tree as your guide you travel

A ?

to mountains

M ?

only to see

C ?

this tree

A ?

before 3.

B ?

Onwards you go to a valley that's mirrored

X ? (V mirrored, as if above a lake)

after a quick break for some tea.

T ?

Amazed by your eyes

I ?

with this valley

V ?

a new twist on 3 is just what you see.

S ?

Frustrated at least, this tree again,

A ?

all of it over, here it ends.

Whew. The only harm actually suffered was by the possible result,
B A D   Y A M C A B X T I V S A,...

...which just can’t be entirely correct.