Having had to get rid of a crab apple tree which was covered in a pwdery white mould which dropped on everything underneath, I was disappointed to find similar white spots on the privet, is everything I now plant in the way of trees doomed?

It sounds like Powdery Mildew which appear on leaves of many shrubs particularly if they are overcrowded and the soil is dry It is common on many shrubs and trees. Spray with a Systemic Fungicide at the first sign of the diseas. Repeat one week later. If not sprayed cut off badly diseased shoots in Autumn. Better to try and kill it then let it run rampant. Make sure they are watered well.

If you are not into chemical sprays then try 1 part full cream milk to 9 parts water as a spray.
There is no need to remove the plants, just give them plenty of light and air and water and the problem will get very much less.