The responsibility falls on the neighbour to trim plants, shrubs and trees which overhang the boundary. They have the right to trim the clematis back to the boundary.
You cannot go onto the neighbour property to do this work without their permission. You may also need the landlord’s permission depending on the terms of their rental agreement. Take care over this as doing this without permission could constitute trespass.
Clematis Montana is lovely but vigorous. It needs to be pruned at the correct time. It flowers on old wood from the previous year, so pruning after flowering ensures you don't cut off the buds for next year's blooms. Every few years they benefit from a hard prune to untangle branches and control the size.
Unless you want someone hacking at it at the wrong time of year, I would ask permission to do it yourself.
If the brambles are coming from that side, the reverse applies. You can only trim back to your boundary but unless the neighbour is keen on having the brambles there for the fruit yield, to feed the birds or for security, I doubt they would object to a very hard prune of that.
