For a customer of mine I washed the seat in the washing machine using the sport clothes program but with cold water, and with added disinfectant. Then I put the whole buggy in the bathtub and washed it by hand, using at first damp antibacterial wipes and the hoover to remove hair and stuff and then just showered it with warm water. I dried with a tea towel. Not easy done as said, but it worked quite nicely, it took me about 40 / 50 minutes all together and the result was that it looked as new! I guess it may require specialist care if you have tore / damaged parts or if you attempted to wash the seat and then put it in the tumble dryer as the fabric may shrink but otherwise it is not an impossible mission.
Joy of Cleaning