Random walk in a switching scenery

For each $x in mathbf{Z}$ let $(eta_t(x))_{tgeq0}$ denote independent copies of a process $(eta_t(0))_{tgeq0}$ defined as follows. The process $eta_t(0)$ takes values in ${-1,1}$, where $-1$ and $1$ denote the values of a scenery with two possible states, and

• $eta_0(0) = -1$ with probability $1/2$ and $1$ otherwise;
• at the arrival times of a Poisson process $N_t(0)$ with rate $lambda > 0$, $eta(0)$ switches state, ie
  begin{equation}
  eta_t(0) = (-1)^{N_t(0)}eta_0(0).
  end{equation}

Let $X = (X_t)_{tgeq0}$ denote an independent continuous-time nearest neighbour random walk on $mathbf{Z}$, ie at rate $1$, $X$ jumps to one of its nearest neighbours chosen with equal probability.

What is the mean first time that $X$ encounters a scenery different from the one in which it started?

That is, denoting
begin{equation}
T := inf{ t > 0 : eta_t(X_t) = – eta_0(X_0) },
end{equation}
what is the value of $mathbf{E}T$?

There are potentially loads of applications but the ones I have in mind are biological. Imagine the diffusing particle represents the motion of an organism through a one-dimensional habitat and $eta_t(x)$ denotes the presence of a virus at $x$ at time $t$. How long until the organism encounters the virus?