Buy a cheap tarpaulin from B&Q for less than £5 and put it over the shed to keep the rain out. Check the timber and replace any rotten pieces. In the spring when it's fine - or sooner if you have the time put on underfelt and then the top felt (with little bits of grit on it) seal the overlaps with cold bitumen paint and bang in some large headed galvanised clout nails. Run a wooden batten along the edges where appropriate. Wear old clothes.
The birds are probably eating bugs in the rotten wood - and just pulling at the felt to get to their meal
Peccavi
October 2008