It's endemic in our allotments too. I took on another half plot (the other half of my half plot) over the winter, and it's worse on it, so having to work hard to keep it down. Unfortunately the next door plot is dormant at present due to illness of holder, so it keeps spreading. I have found in my own plot that there is no easy or short answer, just keep pulling it out and it gets better, but doesn't go away. Not sure what does kill it, probably treating with a powerful weedkiller at roots but that isn't feasible for me.
Like cockroaches, it would probably survive nuclear holocaust.