Here is an answer. If it inspires you to change the question, please don't. Just ask a new one.

No, you can't even say that the  $8$ unknowns are bounded by $T^2-1.$  


To recast, given an integer matrix $A=\begin{pmatrix} 
a & c \\
b & d 
\end{pmatrix}$ with entries bounded in absolute value by $T^2,$ you wonder about finding integer matrices $W=\begin{pmatrix} 
w_1 & y_1 \\
w_2 & y_2 
\end{pmatrix}$ and $X=\begin{pmatrix} 
x_1 & x_2 \\
z_1 & z_2 
\end{pmatrix}$ with smaller entries and $WX=A.$

Since $\det(W)\det(X)=\det(A)$ there are constraints. For example with $T=1000$ and $p=999983$ consider 

$$\begin{pmatrix} 
w_1 & y_1 \\
w_2 & y_2 
\end{pmatrix}\begin{pmatrix} 
x_1 & x_2 \\
z_1 & z_2 
\end{pmatrix}=\begin{pmatrix} 
p & 0 \\
0 & 1 
\end{pmatrix}.$$

As $p$ is prime, one of the two factors has determinant $1$, say the second. Then you can work out that 

$$\begin{pmatrix} 
w_1 & y_1 \\
w_2 & y_2 
\end{pmatrix}=\begin{pmatrix} 
p & 0 \\
0 & 1 
\end{pmatrix}\begin{pmatrix} 
z_2 & -x_2 \\
-z_1 & x_1 
\end{pmatrix}=\begin{pmatrix} 
pz_2 & -px_2 \\
z_1 & x_1 
\end{pmatrix}.$$

———

I wonder what a counting argument would give. There are about $16T^8$ ways to pick the $4$ parameters and about $256T^8$ ways to pick the $8$ unknowns. So a $16:1$ ratio. But that drops to $2:1$ once you factor in that you can swap the roles of the various $u_1,u_2$ and/or negate some pairs.

In many cases there are enormously more solutions .