I think the correct definition should read
UConv[w_] := {Convolve[U[x], w[[1]] /. w[[2]] -> x, x, w[[2]]], w[[2]]}
in order for Nest to give the expected output (confer details above).
Best, Xavier
Your code has some errors. I think you want:
UConv[w_] := {Convolve[U[x], w[[1]], x, w[[2]]], w[[2]]}
Nest[UConv, {U[x], y}, 2]
