Do you have a hose? You could hose off the fleece, that would probably work.
P.S. I kind of can't help but notice that you posted this thread twice. Was it a mistake? I'm pretty sure that you shouldn't post things twice. Sorry if I sound mean or something but I thought that might be good to point out...
The duplicate thread was deleted. Please pay attention to your threads when you post them from now on. If your computer freezes and submits a duplicate, send a private message to a moderator, so we can fix it.
As far as not using the washing machine, you would need to be sweeping up the poops or getting a hand vac for them anyway, so they would not be going in the washer. Guinea pig pee is probably cleaner than human considering their diet, activity level, and lack of exposure to outside influences, so there is really no way that washing the fleece in your washer would get you sick. I don't really see a difference in washing fleece and towels (if you use them) from the pigs, and washing human underwear, baby clothing, or human clothing if you should get sick or have an accident. Detergent and hot water is going to kill any germs from your guinea pig.
As PM said, the fleece will not make you sick. But if your mum won't let you use it, you could take it to a laundrette. Or if it comes down to it, you could try washing it in the bath by hand. Make sure you've swept all the poos and hay and hair off first.
If none of these are possible, and you haven't yet got the fleece, maybe you should think about keeping with your usual bedding?