Tea tree oil is great for removing mould on hard surfaces, so it just might work on roller blinds. (If nothing else, it smells better than bleach!) Try adding about 15-20 drops to a bathful of water and soak them, or use a smaller amount of water with a few drops of oil in it to treat spots on the blinds.
By the way, what is it with people not ventilating their houses properly?? I had the very same problem as you when I moved into my house. The blinds in the master bedroom and en suite were foul. They went straight into the skip! We suffered bare windows for a while, then eventually managed to buy some net curtains.