= Solution
If $\delta$ is the cross-stream transition width and $\Delta U$ its speed change, dominant horizontal shear requires
$$
\frac{\Delta U}{\delta}\gg |u_z|.
$$
Balancing the two terms in the reduced ODE gives the scale
$$
\delta^{n+1}\sim\frac{HKh}{\mu}(\Delta U)^{n-1}.
$$
The approximation is strongest in the margin where $U'$ is large, for a thin basal layer $h\ll H$ and parameters making this lateral scale short compared with the scale on which vertical shear changes the plug velocity. It fails sufficiently far into either uniform region because $U'\to0$ while the $O(\epsilon)$ vertical shear needed to transmit the driving stress remains. It can also fail in narrow neighborhoods where neglected free-surface, bed-transition, or three-dimensional effects vary on the same scale.
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