Injured Wood Pigeon

Hi. On Monday evening whilst walking the dogs we found a wood pigeon in bushes by the road. He was just sitting there. When I approached he fluttered along the ground dragging both legs. He managed to fly up into a tree, but couldnt perch and just came crashing back down, so we picked him up and brought him home. There was no blood on him anywhere and no obvious sign of any injury. He appears to be an adult as he has the white neck patches.There doesnt feel to be any breaks anywhere on his legs, but they are both limp, with no tone at all and no grip in his feet. I rang the local wildlife rescue, who said that Woodies dont do well in rescue and advised us to take him to the nearest vet for euthanasia. As he doesnt appear unwell or distressed we were reluctant to do this. He is at present living in our shed. He appears to be eating (mixed corn and oystershell grit) and I'm presuming he is drinking. He appears only to have done one poo. He sits quite calmly until we go in, when he flutters into a corner. Both legs are still limp and dragging. Suggestions please on care as we dont really know what to do next- or a rescue center in Shropshire/West Midlands that may be interested in taking him.