Elbow grease and poison, I'm afraid - you need to remove all the top growth, dig the soil and take out as much root material as you can, then those that are left (there will be plenty) that you can't dig out, drill or cut into them and apply SBK, a brushwood killer (don't spill it or get it on your skin). Cover the treated areas and wait - you will probably have to dig again and re-treat those roots you've missed.

Often, when you visit a 'bluebell wood', the reason the bluebells are not smothered by brambles is that there are trees growing that shade them out at the vital times of year. Of course it depends on the scale of your project, but a sustainable solution might be to plant beech or oak trees?