= Solution
Let $R=|U|$. Uniform volume measure on the unit ball gives $\mathbb P(R\leq r)=r^n$, $0\leq r\leq1$, hence radius density $nr^{n-1}$. Conditional on $R=r>0$, <independence> leaves $W$ a standard <Brownian motion>. Its exit time from the radius-$r$ ball is finite: applying <optional stopping> to $|W_t|^2-nt$ stopped at that exit and then letting $t\to\infty$ gives $\mathbb E\tau_r=r^2/n$.
The <probability law> of the exit point is invariant under every <orthogonal transformation>, because both <Brownian motion> and the centred sphere are invariant under those transformations. The unique invariant probability on that sphere is normalized surface measure; for $n=1$ it is the equally weighted pair $\{-r,r\}$. This is also the conditional angular law of $U$ given $|U|=r$. Mixing these identical conditional laws against $nr^{n-1}dr$ proves the <uniform ball sampling by Brownian stopping> identity
$$
\boxed{W_T\stackrel{d}=U.}
$$
The event $R=0$ has probability zero and causes no difficulty. <Independence> of $R$ permits its value to be included in the initial <filtration>, making $T$ a <stopping time>.
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