You need to download the registered plan of your property from the Land Registry Office (around £4). Hopefully the responsibility for the boundaries will be shown by T markings.
Large Ts should be printed along the boundary, on their side. If the leg of the T is in your garden, you are responsible, if it is in their garden, they are. If there are no Ts but there are Hs it will be shared responsibility.
Be aware that having responsibility does not mean you have to erect a fence, wall, hedge or any other partition. It means that any partition must be safe from causing damage to the neighbours or their property. It is perfectly legal to take down the partition and not replace it.
You can erect a partition on your side of the boundary and they can do what they like their side.
If it is their responsibility and they do erect a partition, you cannot paint it, bang nails in or do anything else to it without your neighbour's consent. And vice versa.
So, if it is their responsibility as you believe it is, they do not have to put anything up at all.