Want to take the vine down!!

I believe you can as long as you give the clippings back to your neighbour. In the interest of good neighbourliness, I would inform them what you are going to do.

If the land in which it's planted is right alongside your house, it's more difficult. Is your property simply the wall? There are often quite complicated agreements about access to walls when you need to stand on your neighbours' property to maintain your wall.
The legal situation is as Cammomile describes, that you can remove anything within your property but you must return it to the 'owner', even if it has no value as cut branches or stems.
Discuss it with your neighbour but why do want to remove it? Is it unsightly or doing damage to your property?

Thank you,
We want to take it down because we don't like it and are worried that it will damage our house as its now starting to grow really high.
We will have a chat with the neighbours first but just wanted to know where we stand if they said no.

Just to confirm clearly, anything which is rooted in next door's garden which crosses the boundary into your territory can be removed - though the clippings do belong to the neighbour, as stated above. Climbers should be grown up a support which belongs entirely to the person growing it, so not a shared wall, or shared fence - there should be a separate support for any climber. Hope that's clear enough;-)

Try to remain amicable with your neighbour - I work with solicitors and I've seen issues like this blow up out of all proportion and end up costing both parties a lot of money in legal fees. Don't simply go onto your neighbours property and make a mess without their permission!!

Interesting to hear Sid's comments, as I remember meeting an old chap once who had had an argument with his neighbour about exactly where their boundary was in relation to a hedge which had been planted. He'd already spent around £5000 on legal fees and his neighbour likewise. And the dispute still hadn't been resolved as there were no clear plans which showed the precise location of the boundary.
Arguments just aren't worth having.

I realise I wasn't clear enough - you are not within your rights to enter your neighbour's property, nor to remove from his territory a plant growing there; you are only allowed to cut back what is invading your own territory.