I would leave the nest where it is, wouldn't move it if there isn't a good reason to do so. Putting it into a nestbox or a low barrier around it might be a good idea though, it reduces the risk that kits are dragged out at the teat or that they crawl out.
First thing to do is to remove the buck, second thing is to mark 4 weeks after giving birth in the calender - unfortunatly to rabbits there's no aphrodisiac like a doe giving birth, she might be pregnant again already.
Meanwhile, think about how to contain the kits once they become mobile - you don't want them free roam. They pee and poop everywhere, and worse, they get under and behind everything.
I never had any issue with other does around, they respected the nest, I think one even started to lactate and nurse. Only problem could be if they are pregnant too and pick the same spot for a nest. My breeding does live in pairs (or trios at times), but a few years back there was an accidential litter in a hutch with 4 doelings - didn't notze anything until there were 5 more rabbits hopping around in that hutch.