= Solution
The Markov factorization is
$$
p(\mathbf y\mid\mathbf t,\mu,\tau)
=\varphi(y_1;\mu,R)
\prod_{j=2}^3
\varphi\!\left(y_j;\mu+\rho(y_{j-1}-\mu),R(1-\rho^2)\right).
$$
Differentiating its log-likelihood with respect to $\mu$ gives
$$
\widehat\mu
=\frac{y_1+(1-\rho)y_2+y_3}{3-\rho}.
$$
Its coefficients sum to one, so it is unbiased. Direct covariance calculation, equivalently inversion of its <Fisher information>, gives
$$
\operatorname{Var}(\widehat\mu)
=R\frac{1+\rho}{3-\rho}.
$$
Thus the variance tends to $R$ as $\Delta t/\tau\to0$, because the observations become perfectly correlated, and to $R/3$ as $\Delta t/\tau\to\infty$, because they become three independent draws.
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