Hi, I am with a large disaster recovery contractor. I have dried historic wood items that were submerged in river water for weeks. This includes museums with cultural artifacts, such as chairs, spinning wheels,etc. and building items such as wooden moldings, trim etc. The key is a gradual reduction of ambient humidity over a period of weeks. The spokes should be restrained with something rigid such as rigid aluminum to create jigs that will not allow the spokes to warp during the drying process. The wheel should have clamps installed to clamp the wood to the steel outer rim. The steel outer rim would in effect become the jig. We have freee drying chambers but i would not recommend them.
You should try to remove the easy to remove sludge and debris while the wheel is in the water. Cleaning with ph neutral surfactants can help remove oils that may be clinging to the wheel. Change the water a few times.
Others can comment on the restoratoin and preservation of the wood once it is out. Please call if i can provide further advice.