Good idea Kid :)
IME very few independent rescues or sanctuaries have on site vets so an injured animal will need treatment first.
WRT dogs, you are legally obliged to inform the dog warden and I'd advise the police too although they are no longer responsible for strays unless they come under the Dangerous Dogs Act. You can still report the dog as found and ask the police to note it as they would an item of property, in the hope that the owner has reported the dog as lost. You may keep the dog once you have reported him to the DW provided that they have no reason to believe that you will neglect or harm him, you agree to keep him for 28 days in the first instance and return him to the owner, making all reasonable efforts to locate the owner therefore.
In practice a kind rescue MAY take in a stray you have found and report it to the DW and gain their permission to keep him although I guess strictly speaking you shouldn't as the duties lie with you, the finder, not the rescue and he shouldn't really be passed on to a third party. However, I've known this to happen and be approved by the DW. Never assume that a rescue WILL though as they may be full and/or not have a good relationship with the DW, so call ahead before turning up!
Sometimes, as I said earlier, a vet will have good contacts in rescue, both larger ones and daft bints like me, and know where to send an injured animal on to for sanctuary/rehabilitation/further care. Always worth asking what your vet's policy and contacts with rescuers is like when you next visit.
I still find myself surprised at the number of rescues out there for all kinds of animals and birds, and am still learning of new individuals who will take in specific species. I've met ladies who take in hedgehogs and ducks, have dealt with and been helped by lovely sanctuaries when I've found and reared abandoned pigeons, had a parrot rescue pick up and bring to me a pound dog who was about to be PTS, all sorts. If it helps anyone I can always put out a shout to see if I can find help in such situations.
Often it's as much about who you know as what you know, about who you may NOT know but who someone else in rescue circles does know and having, in my case, your name known as a foot in the door when the rescue is struggling for space and might otherwise regretfully turn you away.