I enjoy removing brambles from hedges as a Winter task
I leave them where I can, but when they are a nuisance by a footpath then I remove them each Winter, saves having to untangle 20 foot bramble monsters in the summer heat
In Winter there are minimal or no leaves in the deciduous hedges, easy to see the brambles and their thorns
Long armed welders gauntlets are less than £10 online and just as effective as the horticultural variety
use a trowel, spade or a hori hori to dig out and pull most of the roots up
Keep it up gradually each year and the problem will be removed