`Reduce` with an inequality that involves `ProductLog` is running forever

Try this

Do[
  lm=RandomReal[{1,10}];bD=RandomReal[{-10,10}];bL=RandomReal[{bD,10}];
  bmax=RandomReal[{bL,10}];W=RandomReal[{0,10}];WB=RandomReal[{0,10}];
  If[bmax((lm-1)W-WB)>lm W bD,
    If[Im[bmax E^ProductLog[(bD W+bmax WB)/(bmax W-bmax lm W)]]!=0,
      Print[{bmax E^ProductLog[(bD W+bmax WB)/(bmax W-bmax lm W)],">",bD,
        bmax((lm-1)W-WB)>lm W bD && bmax>bL>bD && lm>1 && bmax>0 && W>0 && WB>0,
        "bmax=",bmax,"lm=",lm,"W=",W,"WB=",WB,"bD=",bD,"bL=",bL}]
]],{16}]

which after some delay will print things like

{2.21858+4.86767 I,>,-1.67293,
 True,
 bmax=,9.00708,lm=,2.08953,W=,2.69263,WB=,2.68902,bD=,-1.67293,bL=,1.44401}

showing that for some values of your parameters which satisfy all your assertions that your expression is not greater than bD because your expression is complex. True shows that all the assertions were True. The list of parameter names and values shows the exact combination that results in this counterexample.

If it doesn't print anything then run it again because sometimes 16 iterations isn't enough to find a counterexample.

Check all this VERY carefully to make certain there are no mistakes.