Wald’s identity for Brownian motion with $E[\sqrt T]<\infty$.

Since you have shown $W_{T\wedge t} \to W_T$ in $L^1$, one has $W_{T\wedge t} = E[W_T \, | \, \mathscr F_t]$, and so by Jensen's inequality,

$$W_{T\wedge t}^2 \le E[W_T^2 \, | \, \mathscr F_t].$$

Taking expectation and letting $t\to\infty$, one sees

$$ \limsup_{t\to\infty} E[W_{T\wedge t}^2] \le E[W_T^2].$$

Applying Fatou's lemma gives the complementary inequality, so $E[W_{T\wedge t}^2] \to E[W_T^2]$ as $t\to\infty$. Since $E[T\wedge t] \to E[T]$ by the monotone convergence theorem, the result follows.