  In this paper we extend to $UCW$-hyperbolic spaces the quantitative
asymptotic regularity results for the alternating Halpern-Mann iteration
obtained by Dinis and the second author for CAT(0) spaces. These results are
new even for uniformly convex normed spaces. Furthermore, for a particular
choice of the parameter sequences, we compute linear rates of asymptotic
regularity in $W$-hyperbolic spaces and quadratic rates of $T$- and
$U$-asymptotic regularity in CAT(0) spaces.
