I don't think its worth the risk, frankly - its possible the honey fungus is already present inside the roots or lower stems of even the healthy plants, so even if you wash off the soil, you risk spreading the fungus to another area. Privet is very susceptible to the nastier forms of honey fungus - you'd be best off choosing a resistant variety of plant instead. More information here https://www.rhs.org.uk/advice/profile?PID=180

Mark is planning a fence rather than another hedge. It has been known for the fungus to attack wooden fences, so it might be good plan to use either concrete uprights or metposts?

I agree I wouldn't transplant them. I had honey fungus that slowly killed a privet hedge. I dug out the affected plants and then the plants 2m either side of the diseased ones. the others survived [well 10yrs on they are still alive]. the wooden fence posts were destroyed as stera said. so the replacement were metapost uprights.

welcome to GoY too :o)