That's a tough one. You can't poison the ground because it would inevitably end up in the brook, and poisoning a waterway is an offence, and something to be avoided for environmental reasons anyway.
Spraying them with something like Roundup won't kill them - it'll check them a bit, but it won't kill them off. Who owns the land behind your fence?

You could prune them back to the fence line and drop the prunings over the fence

Thanks for the answers, as for who owns the land directly behind the fence, we do, we own half way into the brook, sounds nice, if it was an attractive one, but at this point it isn't, and as it's a steep bank down and the brambles have been there for years it looks like it will be an industrial sized job, and expensive to dig them all out.

As for cutting them back, it's what we do every year, but as you can imagine it's a constant struggle and we are getting fed up.

Could you perhaps treat them like a hedge and cut them back with hedge trimmers before they overhang very far?
Digging them out is very discouraging as they grow back from any bits left behind. If you can reach with a trimmer or brushcutter regular cutting to the base will eventually get rid but it does have to be regular and cut right to the ground.

As it seems you have no interest in growing on the bank what is the problem with just cutting back?

I understand why Maxine wants rid of them - they're sneaky blighters, creeping through cracks in the fence or underneath and rooting before you notice.
The only answer is to get over there, risk life and limb, cut them all down, dig up to expose the knobbly roots, drill into them, making small holes, then fill the holes with SBK, cover with upturned pots or something, and leave the poison to do its work.

Thanks for all the help. My hubby has a few days off at Easter and is going to see if he can at least make a start.