I don't think there's one right
answer:
1. If your kids can do without the swing, you could try "bagging" the lower part of the swing, but leave the chain out. Before doing so, choose a place in the woods where the nest is less likely to be a problem and ideally a high
enough branch that the swing seat will be well above head height. That really means that you've got to choose a spot and put a ladder up before bagging the swing. Then you need to carry the bagged swing as gently as possible to its new summer location. Then I suspect the best way to get the bag off without being stung is to not do so... just use a knife to quickly slice a decent hole in the bottom. The wasps shouldn't have any trouble finding their way out, but hopefully won't do so until you're at a safe distance.
2. My humane way of killing wasps if the nest is small is to go out after dark with a paper bag and a scraper if the nest is attached well. I close the bag around the nest, and then just put it on a flat surface and stomp on it. Not fancy, but quick and permanent.
I have lots of wasps around my well shed area where many of my annuals are planted because that's were the sun is best. However, they're not aggressive. I've read that wasps can learn to identify different humans, and if you can convince them you're friend, not foe, they won't sting you. I'm not convinced that works with some of the more aggressive/territorial wasps. I am not allergic to the sting, but I have about the strongest negative reaction that's possible without being allergic. So I'll back up the people who say the nest can't stay where it is. I know to move slowly and deliberately in the area, but children are children and stings can be more dangerous for some people than others.