Well talk to your neighbour about their ivy, you can cut back wahatever comes through to you but shouldn't put weed killer on it. The stuff on your side, cut through the trunk low down and then after it has wilted, pull off what is on the walls then use a glyphosate weed killer on the new shoots DON'T be tempted to cut or pull them off, justkeep on spraying the glyphosate as it regrows, you will need to be patient, maybe take a year or two but it will work eventually
I sympathise...ivy is the devil to get rid of or even control