  A recently-introduced class of probabilistic (uncertainty-aware) solvers for
ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to
initial value problems. These methods model the true solution $x$ and its first
$q$ derivatives \emph{a priori} as a Gauss--Markov process $\boldsymbol{X}$,
which is then iteratively conditioned on information about $\dot{x}$. This
article establishes worst-case local convergence rates of order $q+1$ for a
wide range of versions of this Gaussian ODE filter, as well as global
convergence rates of order $q$ in the case of $q=1$ and an integrated Brownian
motion prior, and analyses how inaccurate information on $\dot{x}$ coming from
approximate evaluations of $f$ affects these rates. Moreover, we show that, in
the globally convergent case, the posterior credible intervals are well
calibrated in the sense that they globally contract at the same rate as the
truncation error. We illustrate these theoretical results by numerical
experiments which might indicate their generalizability to $q \in
\{2,3,\dots\}$.
