Personally I would wait and make sure all of your weeds have gone before planting anything else. The confiers may have died as they are greedy plants, and will take all of the moisture and nutrients from the soil. If you have poor soil it may account for the conifers not surviving.
You need to know if the plant that is rife in your neighbour's gardens is something they've cultivated, it does look like ivy (I think the leaves are too big for it to be bindweed). You don't want to get off on the wrong foot witlh the neighbours by killing anything they're culitvating!
If the neighbours regard it as a pest, then I'd use weedkiller to try and get rid of it, a weed is just a plant in the wrong place. If it's something that they've cultivated, do not use weedkiller on it, just cut any off that comes over on to your side, and put it in the green bin if you have one, don't put it in the compost bin, it will just sprerad around your garden when you use the compost!
You need to know what type of soil you have, and how rich it is before planting anything else, otherwise you may well just be wasting your money. The soil will need enriching to put back any nutrients taken away by the conifers. By all means, remove the conifers, but you may want to think about putting in a fence rather than a hedge, until you have sorted out the soil. Please don't just take out the conifers and leave no boundary - good fences make for good neighbours.