= Solution
Scale axial distance with $\lambda\hat h$ and normal <velocity> with $\hat U/\lambda$, while the dimensionless axial <velocity> has scale $\hat U$. Let $r=R+y$; keep $R>0$ fixed on the asymptotic scales. In the core, the ratio of transverse viscous terms to axial convective terms is of order $\lambda/Re$. A small indentation and the associated <displacement> create tangential slip of order $\epsilon$ at the edges of the wall layers.
For a layer of relative thickness $\delta\ll\epsilon$, that slip makes axial convection of order $\epsilon^2/\lambda$, while transverse viscous diffusion is of order $\epsilon/(Re\,\delta^2)$. Balancing them gives
$$
\delta\sim\left(\frac{\lambda}{Re\,\epsilon}\right)^{1/2},\qquad \delta\ll\epsilon\quad\Longleftrightarrow\quad Re\gg\lambda\epsilon^{-3}.
$$
Unsteady diffusion gives the equivalent estimate $\delta\sim(Re\,St)^{-1/2}$ when $\lambda St=O(\epsilon)$. Thus a consistent regime supporting thin <viscous boundary layers> and an <inviscid core> is
$$
\boxed{\epsilon\ll1,\qquad\lambda\gg1,\qquad Re\gg\lambda\epsilon^{-3},\qquad\lambda St=O(\epsilon)}.
$$
At a location where both imposed unsteadiness and slip vanish, a shear-dominated entrance balance can instead have thickness $(\lambda/Re)^{1/3}$; the same strict Reynolds-number condition keeps even that scale below $\epsilon$. The reduction below assumes that no wall-layer thickening invalidates the stipulated small thickness. The normal flux displaced by a slip layer is of order $\epsilon\delta$, smaller than the retained $\epsilon^2$ core flux. Tangential no slip is imposed within these layers, not directly on the leading inviscid core.
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