I have one large flower in my garden that gets repeatedly overcome with convolvulus

Digging everything out and extracting roots won't solve the problem, and there's nothing you can use that will work which will still allow you to plant anything else for a year or more. If its next door, it'll be the gift that keeps on giving unfortunately. You can either just keep pulling it out wherever it grows, or put bamboo canes in where its growing, let it get to the top, then spray thoroughly with glyphosate. I know you've tried treating the leaves before, but allowing it to climb a cane, then hitting it with herbicide works better, preferably while its still growing strongly, not when its in drought. It won't entirely get rid of it, but it will reduce the amount if you're rigorous and vigilant enough.

Poisoning the whole bed to any depth wouldn't be a viable consideration. I would say that even if you clear the whole bed and be prepared to leave it for a whole year; digging it over many times as the bindweed re-emerges you might not clear it up for good. I would continue to deal with it as it emerges with glyphosate mixed with wallpaper paste and applied with a glove. This, unfortunately would be an on-going battle. As far as the bindweed emerging from your neighbour; you would need to make a barrier with something like 'Rootguard'.

Thank you both so much, it has given me a better plan of action. Needed pointing in the right direction. Cheers x