Let's call a tree front-row iff there are no other trees directly between you and it. Clearly any non-front-row trees are covered by front-row trees.
We call a pair of front-row trees adjacent iff, from your perspective, no other tree's midpoint appears between the midpoints of those two trees.
Claim: You cannot see through the gap between any two adjacent front-row trees.
If this is true, then clearly you cannot see out of the forest, since any sightline passes between some two adjacent front-row trees.
Proof of claim: Let's take any two adjacent front-row trees $A$ and $B$.
Suppose you can see between them. Then you can see their midpoint $C$, since the line $UC$ (pun intended) has equal distance to $A$ and $B$.
We know that the distance $|UC|$ is greater than $250m$, since otherwise there would be a tree at the integer coordinates $A + B = 2C$ in contradiction to the adjacency of $A$ and $B$.
Now let's look at the area of $UAB$. By Pick's Theorem, we know that $UAB$ has $0.5$ times the area of a grid square, so in this case the area of $UAB$ is $50m^2$.
Now consider the triangles $UCA$ and $UCB$ which make up $UAB$. They each have a base of over $250m$ and a height of at least $20cm$, so they must each be over $25m^2$ large. Together they must therefore have an area of strictly more than $50m^2$, which is clearly a contradiction to the previous paragraph. This proves the claim.