Good that you asked because I recommend using a sharp spade to cut all bramble stems a little way below the surface. Often there are less than it looks, making them much easier to clear than other perennial weeds. A few main clumps will need a deeper cut than smaller peripheral plants which root off branches falling onto soil. There is no need to remove all the small roots, just the main stem, to the depth at which it divides into many small roots. If you cut down the tops and mulch them, re-growth will be abundant!
Every starting point is different and needs careful assessment. No dig is not always the best way to begin, but in this case you will not be digging over the whole area, although it does depend what other weeds there are. If you have couch grass as well, I would mulch for the rest of the year, after brambles are out.