We did this years ago. We cut down the hedge to ground level and then used a stump killer on each of the stumps. There are a lot of stumps because each privet is a shrub with several. None of the stumps ever re-grew and after a year or two we could pull the dead stumps out by hand.

Same answer from me as your question about the hawthorn hedge.

Could you not renovate this hedge too, both your privet and hawthorn can be brought back to there former glory by doing this, are you replacing hedging for fencing?

Privet isn't all that deep rooted, though a mature hedge's roots will probably be quite thick. We found digging out what will come and sawing through what was left was effective and it didn't regrow. Hawthorn and privet are both very good hedges for wildlife though and can be kept under control quite easily.

When I removed our privet hedge and replaced it with a fence I gained a six feet wide strip of garden! In this case I would prefer to remove the privet and try to retain the hawthorn hedge.