I may have missed something but the OP's brambles originate and are rooted on neighbouring property so digging out won't be viable unless they 'trespass'.
I suggested that they ask the neighbour's permission to remove the brambles, Dave. I have found that the sort of people who have brambles are often unwilling to do the job themselves, but perfectly happy to let someone else do it.
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I've cleared brambles without weedkiller too. I found they rooted close to the surface here and they'd hit clay within a few inches.
I may have missed something but the OP's brambles originate and are rooted on neighbouring property so digging out won't be viable unless they 'trespass'.
I suggested that they ask the neighbour's permission to remove the brambles, Dave. I have found that the sort of people who have brambles are often unwilling to do the job themselves, but perfectly happy to let someone else do it.