Is what I've been doing today.
Of course, take off and nuke the entire site from orbit is the usual cure for brambles - it's the only way to be sure. But, in this case I want to rescue the shrubs and hedging through and over which the brambles are entwined and spread.
So, fully gauntletted, I've been reaching in and cutting as far down the stem as I can stretch. Repeat for each stem. Repeat for each sub-stem. Repeat 'cos I missed some. Etc. Etc. It's hot, thorny, and slow work.
Can anyone reassure me that this is going to work? I'm expecting to have to rinse and repeat every few months until the bramble get the hint and die. But - they will die eventually, won't they?