Brick Built Shed, Both timber and Felt in roof need replacing

I have a small brick built shed in my back garden which is linked to my neighbours, the walls forming the back corner wall. We share the same felt roof, and while his side is maintained, mine is a complete mess. The timber beneath the felt is rotted and splitting away, the felt is wrecked. The original timber door came away a long time ago as would not close. I want to keep the original brick / frame as much as possible and somehow get the timber and felt repaired, and fit a new door. What kind of tradesperson can help with this? How can I do it in the most cost effective way?