Damn things! I dismantled our dining room fire twice in a two month period to release woodpigeons which had fallen down the chimney.
The first time I deliberated for a whole day whether to allow it to die or to call out a registered gas fitter at vast expense. In the end I took the fire apart myself, taking great care to remember which part went where. The second time I knew exactly how to go about it and did the job within 20 minutes. Slightly regretted doing it so quickly, as the bird was considerably more active than the first one. A word or two of advice: arrange some netting around the fire area so that a sooty pigeon can't escape and make redecoration necessary (I actually had the sense to do that both times!), and try to grab it (gently) round the body, clamping the wings to its sides.
Don't grab the tail, as the whole lot will come away in your hand! This happened to the first one I rescued; I felt sorry for it, held it while it drank its fill, as it was very thirsty after a day in the chimney, then released it. Twenty minutes later it was back on top of the chimney and cooing down it. That was when I decided to have a ventilated cap put on each chimney!
Woodpigeons seem to like using chimneys to amplify their amatory cooing, and occasionally lean too far and overbalance - well, that's my hypothesis.
Best of luck with yours; please don't cook it alive, and I doubt that it will be able to fly out of its own accord.
in it if you can get it out. You are free to do what you want with it after 
