Born-Oppenheimer approximation
Atkins did a good job on explaining what parametric dependence of the electronic wave function (and electronic energy) on nuclear coordinates is, but failed to mention that in the derivation of the Born-Oppenheimer approximation this dependence is assumed to be continuous and differentiable and that both first and second derivatives of this quantities with respect to nuclear coordinates are in general non-zero.
In particular, for the system described in the text, we have
$$
  \frac{\partial \psi}{\partial Z_j} \neq 0 \, , \quad
  \frac{\partial^2 \psi}{\partial Z_j^2} \neq 0\, .
$$

Now, when the solution of the form (8.3) is substituted into the Schrödinger equation (8.2), the term involving $T_\mathrm{e}$ is trivial: since the nuclear wave function $\chi$ is not a function of electronic coordinates it is just a constant when differentiating with respect to them, so we get
$$
  T_\mathrm{e} (\psi \chi) = \chi T_\mathrm{e} \psi \, .
$$
But the term involving $T_\mathrm{N}$ does not trivially transform in a similar way,
$$
  T_\mathrm{N} (\psi \chi) \neq \psi T_\mathrm{N} \chi
$$
since both $\psi$ and $\chi$ depend on the nuclear coordinates. Rather, applying the product rule twice, we get
\begin{align}
  \frac{\partial^2}{\partial Z_j^2} (\psi \chi)
  &amp;=
  \frac{\partial}{\partial Z_j} \left( \frac{\partial}{\partial Z_j} (\psi \chi) \right) \\
  &amp;=
  \frac{\partial}{\partial Z_j} \left( \psi \frac{\partial \chi}{\partial Z_j} + \chi \frac{\partial \psi}{\partial Z_j} \right) \\
  &amp;=
  \psi \frac{\partial^2 \chi}{\partial Z_j^2}
  +
  2 \frac{\partial \psi}{\partial Z_j} \frac{\partial \chi}{\partial Z_j} 
  +
  \chi \frac{\partial^2 \psi}{\partial Z_j^2} \, ,
\end{align}
so that 
$$
  T_\mathrm{N} (\psi \chi)
  =
  - \psi \sum\limits_{j=1,2} \frac{\hbar^2}{2 m_j} \frac{\partial^2 \chi}{\partial Z_j^2}
  -
  \sum\limits_{j=1,2} \frac{\hbar^2}{2 m_j} \left(
  2 \frac{\partial \psi}{\partial Z_j} \frac{\partial \chi}{\partial Z_j} 
  +
  \chi \frac{\partial^2 \psi}{\partial Z_j^2} \right) \, ,
$$
where the first term is nothing but $\psi T_\mathrm{N} \chi$ and the remainings are designated as $W$.