Hi Camilo,
I'm beginning to see why you wanted NestList; is it so that you can see the progression of the walk?
Here are the individual parts summed into one:
Meander[n_,t_]:=Module[{RND,walk,VonNeumann,initConf,out,m},RND:=Random[Integer,{1,4}];
initConf=ReplacePart[Table[0,{2 n+1},{2 n+1}],RND,{n+1,n+1}];
walk[1,0,0,0,0]:=0;
walk[2,0,0,0,0]:=0;
walk[3,0,0,0,0]:=0;
walk[4,0,0,0,0]:=0;
walk[0,3,0,0,0]:=RND;
walk[0,0,4,0,0]:=RND;
walk[0,0,0,1,0]:=RND;
walk[0,0,0,0,2]:=RND;
walk[0,1,0,0,0]:=0;
walk[0,2,0,0,0]:=0;
walk[0,4,0,0,0]:=0;
walk[0,0,1,0,0]:=0;
walk[0,0,2,0,0]:=0;
walk[0,0,3,0,0]:=0;
walk[0,0,0,2,0]:=0;
walk[0,0,0,3,0]:=0;
walk[0,0,0,4,0]:=0;
walk[0,0,0,0,1]:=0;
walk[0,0,0,0,3]:=0;
walk[0,0,0,0,4]:=0;
walk[0,0,0,0,0]:=0;
VonNeumann[func_,lat_]:=MapThread[func,Map[RotateRight[lat,#]&,{{0,0},{1,0},{0,-1},{-1,0},{0,1}}],2];
out=NestList[VonNeumann[walk,#]&,initConf,t];
Sum[out[[m]],{m,1,t+1}]//MatrixForm]
Maybe Sum isn't what you want. Not sure what the numbers mean.
Eric