Fledgling wood pigeon or collar dove. - HELP

Hi all, 

Found a fledgling next to a busy dual-carriageway. No trees, undergrowth or bushes in sight. Pouring with rain and freezing cold so rescued it (only kind thing to do in the circumstance). It was not difficult to pick up although it hopped around a little bit. 

Due to being found on Saturday morning, we've been unable to reach any rescue centers and the ones we've managed to get in touch with today, either aren't too interested or are too far away to collect. (Sadly we have no means to travel to one as no car). 

We've followed general advice read online and in this forum, we got it back, warmed it up and it perked up - a lot. It;s been a happy little chappy, drinking and eating (pecking up food for itself) but this afternoon has become very lethargic and tends to just sleep/sit. We've been keeping it warm without trying to cook it but it's not really doing much now. We really don't want it to die, it's a lovely friendly little thing and deserves a chance. It doesn't do much chirping atall either.

Initially it tried to do some flying but wobbles over when it does. It now seems to have given up on that idea and prefers to trot around the house or sit. It doesn't wobble around when walking, only when it tried to fly, it doesn't seem to either have the hang of it yet or we're worried there is something wrong with it. It does not appear to have any injuries. We've bought it some bird feed which has seeds etc, we've tried it on some small bread lumps. 

We live in a mostly built-up area with barely any outside space and lots of neighbouring cats. We don't want to leave it outside as the neighbour cat has been seen several times savaging birds. 

We would really appreciate some help, ie - How long until it can fly, and therefore be released. How much it should eat/drink. Should we be thinking about force-feeding it/forcing fluids into it.

We've read somewhere that some green-looking droppings means it's starving, however they seem to come and go. Please see the pics and any advice would be amazing!