We have a right triangle $RST$ with sides:

• $RS = 504$
• $ST = 128$
• $TR = 520$

Since $RST$ is a right triangle, check by Pythagoras:
$504^2 + 128^2 = 254016 + 16384 = 270400, \quad \sqrt{270400} = 520 = TR.$
So indeed $TR$ is the hypotenuse.

In the similar triangle $UVW$, angle $W$ corresponds to angle $T$.
$\tan W = \tan T = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{RS}{ST} = \dfrac{504}{128} = \dfrac{63}{16}.$

Thus the answer is $\boxed{\dfrac{63}{16}}$.