If it had been a tree that had fallen and broken your fence, then you would definitely had a claim. As it was brambles then I think you would have a problem trying to lay blame. You neighbours will say that the fence was already weak if is was just the brambles that had caused the damage. I know how aggravating it is but even if you were to take legal action it would cost a fortune and you might not get the right outcome. I would, however try talking to your neighbours in the first instance and perhaps you could come to some agreement.

Are these literally brambles, i.e., wild blackberries that have grown on their own and happen to be leaning against the fence? Or are they proper varieties of blackberry which the neighbours had attached to the fence in some way?

Kat, you're responsibility is either to leave the fence as is or fix it. If you decide to fix it, notify who you think is the responsible party that they are the responsible party and of your intentions to repair together with a request that they kindly reimburse you for the cost, either whole or in part. Only after you fix the fence and you pay for it do you present the the receipts of the cost of materials and labor to your neighbor. If they promised to pay and now refuse, your final recourse is small claims court if the courts claims limit covers your costs. If they refused at the outset you cannot make them pay for something they told you they had no intention to pay for in the first place. There would be no legal action to take other than a lawyer writing a few letters at your expense and to be sure, the lawyer would not go any further than that for they need bigger fish to fry.

Fences between neighbours can be tricky things.
For the benefit of anyone that might be unsure; Just to be clear, the rule generally is that the left hand side as you face from the street is yours & must have the fair (best) side facing your neighbour.