= Solution
The scale separation makes correlations between distinct observation times negligible, so $K_\theta\simeq A I$. Put $v_i=A+\sigma_i^2$. With a flat prior,
$$
\boxed{\mu\mid y,t,\theta\ \dot\sim\
N\left(\bar\mu,V_\mu\right)},\qquad
V_\mu=\left(\sum_i v_i^{-1}\right)^{-1},\quad
\bar\mu=V_\mu\sum_i\frac{y_i}{v_i}.
$$
The next latent value is likewise approximately independent of the past conditional on $\mu$, with $f_*\mid\mu\sim N(\mu,A)$. Marginalizing $\mu$ gives
$$
\boxed{f_*\mid y,t,\theta\ \dot\sim\ N(\bar\mu,A+V_\mu)}.
$$
When every $\sigma_i=0$, $\bar\mu=\bar y$, $V_\mu=A/N$, and the predictive variance is $A(1+1/N)$.
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