I am in the UK, does anyone know how to get rid of vine weed?

I am not sure what Vine Weed is, Kerry, possibly convolvulus? Anyway, if it is in your flower beds a weedkiller will kill the flowers as well as the weed. Hand weeding would seem to be the only solution.
Can you also 'weed' it out of the field behind your garden - just along your boundary? This will reduce the amount of weed coming through. Our neighbouring farmer has set aside behind our garden and I strim a six foot buffer strip along our boundary to reduce the amount of seed blowing in.

If you mean bindweed (twining habit, goes for miles, white trumpet flowers), there's no permanent, easy solution. Best thing is to pull off everything this year, try to tug gently and remove what roots you can, if your soil is light. Next year, when it starts growing, place bamboo canes or sticks in the ground so it climbs up those, wait till there's a foot or so of growth, then spray with glyphosphate or Round Up. This will kill of some of it back to the roots, but it is pernicious and will return, so regular monitoring is the only answer. You could try this autumn, if possible, to dig the area, removing the roots as you go - they're quite fleshy, white and about a quarter inch thick, but even a tiny fragment left in the soil will regrow. If you get out as much as you can, that will at least reduce the amount growing next year.

I find spraying in late summer is most effective. the glycophospate weedkiller is taken down to the storage areas of the plant and kills off a lot over the winter. if feasible sink a plastic barrier at least 9" deep at your boundary. this will reduce the number of underground stems that invade your garden.

That's interesting Seaburn - here in London, bindweed has a tendency to die back by end of July and isn't growing strongly, so I didn't think it'd have much effect then.

you could use weed killer or pull it out from the roots

thats lucky bamboo, it can be still flowering in september. but as it overwinters the translocated weedkiller certainly does its thing. i have found it a useful strategy on creeping sowthistle too.

The roots go down metres and as bamboo says, spraying is a good way as costant applications weakens the plant over time. Another way is to apply glyphosate to a waterproof glove and run your hand up the garden cane youve let it grow up. This will avoid spray drift onto other plants.
Just dont let any living bits fall onto the soil, it will gorw back from leavs or stems and bin or burn it!

I had a problem with bindweed and my solution was to dig a trench (about 6 or 7 feet deep) along my boundery and place that black weed membrane stuff vertically and then lay it back over towards the fence. I then backfilled the trench and put a small layer of top soil on top of the bit I'd layed over, if that makes sense. Hard work but so far I haven't seen any coming through. :~))