= Solution
The squared chord distance is
$$
\lVert u(t)-u(t')\rVert^2
=2-2\cos\{\omega(t-t')\}
=4\sin^2\left(\frac{\omega(t-t')}{2}\right).
$$
Hence the restricted <Gaussian process> has covariance
$$
\boxed{
k_\theta(t,t')
=A\exp\left[-\frac{2}{l^2}
\sin^2\left\{\frac{\omega(t-t')}{2}\right\}\right]}.
$$
It depends only on $t-t'$, so it is stationary. It is periodic in either argument with period $T=2\pi/\omega$, and Gaussian-process realizations inherit that period almost surely because $f(t+T)-f(t)$ has zero variance.
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