In a previous paper, the first author derived an explicit quantitative version of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behaviour of Mann iterations of
nonexpansive mappings of convex sets $C$ in normed linear spaces.
This quantitative version, which was obtained by a logical
analysis of the ineffective proof given by Borwein, Reich, and
Shafrir, could be used to obtain strong uniform bounds on the
asymptotic regularity of such iterations in the case of bounded
$C$ and even weaker conditions. In this paper, we extend these
results to hyperbolic spaces and directionally nonexpansive
mappings. In particular, we obtain significantly stronger and
more general forms of the main results of a recent paper by W. A.
Kirk with explicit bounds. As a special feature of our approach,
which is based on logical analysis instead of functional
analysis, no functional analytic embeddings are needed to obtain
our uniformity results which contain all previously known results
of this kind as special cases.