I don't know what the rules are where you live, but in the UK, we're allowed to cut back and remove anything growing over our boundaries, but it is illegal to treat, damage or pull out any planting outside of our boundaries, so you might want to check where you stand, since your neighbour may well be the type to cause trouble if you're caught mucking about with their 'plants'.
That said, I'd try a strong salt solution - mix a couple of tablespoonfuls of cooking salt or table salt in 2 or 3 litres of warm water, and water on to the base of the plants nearest your fence, and only those nearest. Then clip back anything that's poking through the fence currently. The salt should kill the nettles you apply it to, and prevent anything else growing for some time, because salt in the soil is toxic to most plants. Otherwise, spray with glyphosate - this should kill the nettles after a few treatments, but won't prevent eventual regrowth. Keep your children away from the area until the spray has dried.
UPDATE (in response to Slim's comment): Yep, you're right, and salt isn't usually ever something I'd recommend except possibly on a small area of waste ground. I have tried it once, just to see what happened, in a planted area, by applying it only to one plant, at the base. It killed that plant, but none of the others was affected, so I think you'd have to pretty much broadcast the salt solution to completely poison the soil in an area. I didn't, though, risk planting that spot again till the following spring, so around 5 months and a lot of rain later.
I also just want to add, I've only said that's what I'd do in these circumstances - what Torben actually does is his choice (just in case any legal issues arise!).