I'm also in the process of de-brambling a very large back garden, I've found what works for me (I also have little ones and am also disabled, so it's a slow process), is to invest in a pair of decent telescopic loppers (it saves bending down), and chop off the stems about an inch from the floor. I stack the green shoots on concrete (so they can't put down roots),and the brown shoots go into a small dustbin-type incinerator and get burnt, together with last week's 'green' shoots, which dry out fairly quickly if they're undercover (an old tarp does the job).
Can some of you experienced composters tell me if there's anything I shouldn't put on a compost heap ( I know cooked food is a no-no), and do I need anything special, or will an old dustbin be OK (space is an issue), or does it have to be a special composting bin? I get lots of veg peelings, what do I need to add to these to make a nice compost rather than a stinky slime?