  This paper deals with the rate of convergence in 1-Wasserstein distance of
the marginal law of a Brownian motion with drift conditioned not to have
reached 0 towards the Yaglom limit of the process. In particular it is shown
that, for a wide class of initial measures including Dirac measures and
probability measures with compact support, the Wasserstein distance decays
asymptotically as 1/t. Likewise, this speed of convergence is recovered for the
convergence of marginal laws conditioned not to be absorbed up to a horizon
time towards the Bessel-$3$ process, when the horizon time tends to infinity.
