  In this paper we apply proof mining techniques to compute, in the setting of
CAT$(\kappa)$ spaces (with $\kappa >0$), effective and highly uniform rates of
asymptotic regularity and metastability for a nonlinear generalization of the
ergodic averages, known as the Halpern iteration. In this way, we obtain a
uniform quantitative version of a nonlinear extension of the classical von
Neumann mean ergodic theorem.
