Do I understand it right that you want fn[z] to work even if z is a symbol (and thus the If function will not be evaluated)?

If so, use With:

Clear[fn]
fn[x_] := With[{ww = ArcTan[x]}, If[x < ww, ww, x]]

fn[z]
(* If[z < ArcTan[z], ArcTan[z], z] *)

See here: https://mathematica.stackexchange.com/questions/559/what-are-the-use-cases-for-different-scoping-constructs

One of the uses of With is injecting evaluated values.

Or perhaps what you want is fn to evaluate only with numbers:

Clear[fn]
fn[x_?NumericQ] := With[{ww = ArcTan[x]}, If[x < ww, ww, x]]

Yes, Piecewise is more suitable here because it is explicitly designed for symbolic computations. If is primarily for flow control when programming, but it tends to work with many symbolic math functions too.

If[condition,t,f] is left unevaluated if condition evaluates to neither True nor False.

I see, that's the reason. So I need to explicitly tell mathematica to evaluate it, e.g.

fn[x_]=Module[{vv,ww},
 ww=ArcTan[x];
 If[x<ww, Evaluate[ww], Evaluate[ (x+ww)/2 ]
]

which yields the desired result

If[x < ArcTan[x], ArcTan[x], (x + ArcTan[x])/2]

Similarly, with a Which statement I need to sprinkle Evaluate in the right places:

fn[x_]=Module[{ww,yy},
 ww=ArcTan[x];
 yy=Exp[x];
 Which[
  x<0, Evaluate[ww],
  Evaluate[x<ww], Evaluate[yy],
  True, Evaluate[2*yy]
 ]
]

which yields the desired result

Which[x < 0, ArcTan[x], x < ArcTan[x], E^x, True, 2*E^x]

Thank you.