Extend the bounding box to include a little outside the region:
\[ScriptCapitalR] =
ImplicitRegion[
Abs[x] <= width/2 && y <= heigth + ydatum &&
y >= x^2/(2 radius) + Aw (1 - Cos[(2 \[Pi] x)/\[Lambda]w]) + ydatum,
{{x, -0.36, 0.36}, {y, -0.02, 0.3}}]; (* <--- N.B. *)
mesh1 = ToElementMesh[\[ScriptCapitalR], "MeshOrder" -> 1,
MaxCellMeasure -> {"Length" -> 0.01},
MeshRefinementFunction -> meshrefine]
mesh1["Wireframe"]
