Bind Weed

Try roundup if you tease it out of the privet and lay it on the ground at the base, dip a paint brush into a tin with a solution of diluted roundup but add a little oil to the round up it helps it to stay on the leaves, as long as you do not touch the privet leaves it will not hurt and once it is in contact with the ground it nutralises. it will take it in through the leaves back to the bind weed roots. Better to brush it on than spray as the spray carries in the breeze to where you don't want it!!! Dont get it on your grass. If you have grass up to the hedge then I suggest you put a piece of plastic down and lay the bind weed on the plastic possably with a wad of newspaper over the plastic to stop any drip off running on the plastic.. It will take a few days before you see any results. And use rubber gloves if you are using the paint brush. seeing as you say it is coming under the wall from the grave yard, I suggest you run a sweeping brush dipped in the roundup along the base of the wall on the bind weed growth, which will take it in through their leaves back to roots.

If the privet is big enough then you can risk spraying the lower parts of the bindweed with roundup.
I once had a huge privet hedge with an infestation of ground elder beneath it. I sprayed and this killed the ground elder but had no effect on the privet. The privet was large; when I removed it a few years later I gained a meter and a half of garden!

I pull as much out as i can during the summer and then about late august/september i apply roundup type weed killer, a systemic one, and this is taken down into the storage stems and it dies during the winter. I have worn rubber gloves with cotton/wool gloves over the top and then wet gloves with weed killer and stroked the weed to coat the leaves without damaging the shrub/plant underneath.