I can totally sympathize, my fiancé is totally opposed to pigs indoors but luckily I have a cosy shed, heat pads and sleeping bags.
I think maybe the softy softly approach is the best way to go with your mum. If they're anything like my parents, if you've got one parent on board then you're a good way to wining the fight eventually! Maybe if you start researching C&C cage designs, you can show your mum some of the amazing pictures of other people's cages and how hygienic and good looking they are. It can be hard to imagine such a stylish guinea pig habitat when your piggies are in a hutch full of hay, but some of the fleece lined cages look so tidy and neat. Plus if anything, your furbabies are more likely to pick up parasites and attract vermin outside as that's where they come from - a large reason why many cat owners won't let them play outside for fear of fleas. If you keep the hay and food inside, it's no more likely to attract things than human food is.
In the meantime, like
@PiggyOwner and
@Alex2016 said you can start putting things in place that will help for now ie heat pads and hutch cover. I think that wouldn't be unreasonable to ask mum that since they can't come in she help provide things to keep them safe outside.
Surely if you're proving to her that you're being a responsible pet owner, she will begin to appreciate that fact and eventually come round to the idea. Good luck!