= Solution
Apply the orthogonal transformation
$$
D_t=\frac{A_t^+-A_t^-}{\sqrt2},
\qquad
C_t=\frac{A_t^++A_t^-}{\sqrt2}.
$$
The processes $D$ and $C$ are independent one-dimensional Brownian motions, with $D_0=\sqrt2a$ and $C_0=0$. The meeting time is the first time $D$ hits zero, which is almost surely finite by one-dimensional Brownian recurrence.
The <Brownian reflection principle> gives the first-passage density from $x>0$ to zero as
$$
\frac{x}{\sqrt{2\pi u^3}}e^{-x^2/(2u)}.
$$
Substituting $x=\sqrt2a$ gives the <meeting time of two independent Brownian motions> density
$$
\boxed{f_T(u)=\frac a{\sqrt{\pi u^3}}e^{-a^2/u}},
\qquad u>0.
$$
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