Wt stops when any of the three circumstances below occurs:

Wt=b>0

Wt=a<0

t=T

Traditional Gambler's Ruin problem focuses only on the T=inf case.

However, I want to know P(W ends up at b) and P(W ends up at a) and of course P(W ends at T)

This is a tough question and you may consider joining the Mensa group if you can answer this question. You are already a candidate for Mensa if you even solve Xt=Wt+ut hitting problem