How to create the waved style for curves in Graphics?

Not a complete answer.

You can make use of the undocumented function Typeset`MakeBoxes mentioned in this post. Here I'll just code waved line and circle as examples:

(* Stolen from Simon's post, notice the tiny modification. *)
SetAttributes[createPrimitive, HoldAll]
createPrimitive[patt_, expr_] := 
 Typeset`MakeBoxes[p : patt, fmt_, Graphics] := 
  With[{e = Cases[expr, Line[_], Infinity]}, 
   Typeset`MakeBoxes[Interpretation[e, p], fmt, Graphics]]

createPrimitive[Waved[a_, f_, pts_: Automatic]@Circle[p : {x0_, y0_} : {0, 0}, r0_: 1], 
 ParametricPlot[{x0 + Cos[t] (r0 + a Sin[f t]), y0 + Sin[t] (r0 + a Sin[f t])}, {t, 0, 
   2 Pi}, PlotPoints -> pts]]

createPrimitive[Waved[a_, f_, pts_: Automatic]@Line[p : {{_, _?NumericQ} ..}], 
 Module[{fx, fy, 
   distance = Prepend[Accumulate@Sqrt[Total@Transpose@((Rest@# - Most@# &@N@p)^2)], 0.], 
   normal}, {fx, fy} = 
   ListInterpolation[#, distance, InterpolationOrder -> 1] & /@ Transpose@N@p; 
  normal = Sqrt[fx'[t]^2 + fy'[t]^2]; 
  ParametricPlot[{fx@t + a Sin[f t] fy'[t]/normal, fy@t - a Sin[f t] fx'[t]/normal}, {t, 
    0, distance[[-1]]}, PlotPoints -> pts]]]

Usage:

Graphics[{Red, Thick, Waved[1/50, 40]@Line[{{1, 0}, {2, 1}, {3, -1}, {4, 0}}], Orange, 
  Waved[1/10, 50, 51]@Circle[{2.5, 0}, 3/2]}]

Remaining Issues

1. The achieved syntax is slightly different from the expected one, not sure if the expected syntax can be achieved with Typeset`MakeBoxes.

2. The waved style is coded separately for every graphics primitive, so creating a complete waved style still requires huge amount of work.

3. ParametricPlot is relatively slow.

4. The wave doesn't look great at corners:

    Graphics[{Red, Thick, Waved[1/10, 40]@Line[{{1, 0}, {2, 1}, {3, 0}}]}]
   

   

It is possible to make ondulations on a BSpline.

Here is a toy example with a closed BSpline ("closed" in order to see the continuity at the ends)

p={{15.7336, -3.557}, {11.1177, -2.53343}, {15.4259, 19.1467},
   {6.60292, 10.5131},{-28.5053, 10.9099}, {-22.7909, -1.35239},
   {-3.22756, -13.0483},{-17.1309, -32.426}, {6.23965, -7.05847},
   {25.0532, -25.0634}};

f = BSplineFunction[p, SplineClosed -> True];  

Show[ParametricPlot[f[x], {x, 0, 1}]]  

absCurv=NDSolveValue[{abcCurv'[x]==Norm[f'[x]],abcCurv[0]==0},{abcCurv},{x,0,1}][[1]];  

length=absCurv[1];

numberOfTurns = 50;

f1[x_]=f[x] - Sin[2 Pi numberOfTurns absCurv[x]/length]  {{0,1}, {-1,0}}.Normalize[f'[x]];

ParametricPlot[f1[x],{x,0,1},PlotPoints-> 1000]

inspiration source 1 (about {{0,1}, {-1,0}}.Normalize[f'[x]])
inspiration source 2 (about absCurv=NDSolveValue[...)