We keep our litter box in the basement; the cat knows that's where it is, etc. so we wouldn't move it now, plus it's under our laundry sink down there with a little mat in case she trails any of the litter stuff out of it. I haven't been cleaning it since I've been pregnant, my mum has.. but sounds alright to clean it once baby's out as long as you always wash your hands thoroughly (I'm sure people do this anyway when they clean up their cat poo!!)
I know my friend lived on her own for a bit while pregnant, she used those yellow gloves you use to clean dishes, but out of necessity only.