Why would $[0,1) \times \eta$ (with lexicographic order topology) not be a manifold for $\eta > \omega_1$? From Wikipedia's entry on the long line:

And if we tried to glue together more than $\omega_1$ copies of
  $[0,1)$, the resulting space would no longer be locally homeomorphic
  to $\mathbb{R}$.

Why?

Every neighbourhood of $\omega_1$ contains uncountably many ordinals, and hence
$$(\alpha,\omega_1)\times [0,1)\tag{1}$$
is not homeomorphic to a subset of $\mathbb{R}$, since it is not second countable (there are uncountably many disjoint open subsets $\{\beta\}\times \left(\frac14,\frac34\right)$, $\alpha < \beta \leqslant \omega_1$, so the space is not second countable). Since every neighbourhood of $\{\omega_1\}\times\{0\}$ contains a set of the form $(1)$, none of its neighbourhoods is homeomorphic to a subset of $\mathbb{R}$.
