What an interesting question!
The simple answer, unfortunately, is that not one single shrub/tree/fence/barrier can hold brambles back.
They are pernicious, invasive, destructive, and their only redeeming feature is the fact that they bear delicious fruit.
Any shrubs that you plant as a barrier to brambles will simply be infiltrated by new growth, which will then root where the tips touch the ground (another charming habit of this plant) and continue to grow, entwined in and around the shrub(s).
Eventually, the shrubs will be swamped by the brambles, and may even die from the competition.
Solid fences and even walls won't stop them: I've seem brambles appearing over a wall at least three feet above head height, and I've seen them appearing at the base of what appeared to be solid garden walls, having tunnelled underneath from next door.
So what solutions can I suggest?
As you'll already know if you've read much of this blog, I frequently talk about clearing brambles, and the importance of cutting them off just an inch or so below soil level, but that's where they are already present.
Things are quite different if you have already cleared the garden, but are troubled by next-door's brambles coming back in to your area. If next door has occupants, it's always worth going round and knocking on their door, asking them to deal with the problem on their side before it gets to you. Do this now, before you get too worked up and angry about it - just ask them, in a friendly manner, if they would be kind enough to deal with the brambles. In about 80% of cases, they have no idea that they are causing a problem, and will willingly deal with it. In about 10% of cases, they are aware but have been too lazy to do it, and your call might just prompt them into doing some work. As for the remainder, well, there is always the chance of getting "Tough" and a door closed in your face, but it doesn't happen often. And at least you will now know where to throw the snails... [That was a JOKE, ok, a JOKE!]
In many ways, it's actually quite easy to keep invaders out, you just have to arm yourself with a spray-bottle of a glyphosate-based weed-killer, to spritz any new
growth - bramble, ivy, ground elder, anything you see - as soon as it
dares to poke a nose over your boundary.
And whatever you do, don't cut or snap off any long growths - just post the loose end back over the fence, then push as much of it as you can back onto their side. If you cut the stems - and this applies to brambles, ivy and many other climbers - they will respond by creating two or more new shoots, all blessed with enormous vigour, so you are only making things worse for yourself. By pushing them back, you are adding weight to their side, and in time they might just fall away from the dividing fence, which would be a bonus.
If you only have something like chickenwire between Them and you, things are much more difficult, although continued vigilance with the glyphosate can work. And at least you can spray a little more than your own boundary - and no, I'm not suggesting you lean over the fence and weedkill as much of their garden as you can reach, I'm suggesting that you can spray through the bottom inch or two of the wire, to try to catch anything rooting very close to the fence.
If you possibly can, I would suggest erecting a solid fence of some kind, leaving enough of a path for you to access this fence all the way along, then planting screening shrubs inside it. You can then walk along the boundary once a month or so, with your spray-bottle, spritzing any new growth.
And if you can always keen this area clear of invaders, it will make it a lot easier to deal with the ones which will invariable self-seed within your garden.
There, hope that helps!