Another vote for generic glyphosate. Follow the instructions with respect to dose/application rate and PPE. There's no benefit to overdosing especially with something as easy as nettles (broad leaf = plenty of absorption). If they've started to seed though, the seedlings, once they have broad leaves, will need spraying again.
Spray on a windless day. Use a rain-radar website to determine if there is rain on the way in the next few hours - if there is, don't bother. This weed killer works by being absorbed through leaves, not roots. It takes a week or so to appear to start working as it starves the plant of food. If you're deprived of food, you can last a while too
Thought that picture was taken from my back garden for a moment then! We had the same problem when we moved in nearly 3 years ago. I asked the farmer if I could cut them back occasionally and he said no problem as he doesn't harvest that close to the fence. Out came the lawn mower and they were cut as close to the ground as possible. Next two layers of weed control and then some cheap false turf was laid over the lot. Didn't 'professionally lay the turf but made sure edges were tight and joints staggered a bit. Now it looks as though the grass continues into the field and 4' out there is a grass border that one of the farm hands comes and trims down 3-4 times during the summer. Cooperation and happiness all round.
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