Eucalyptus gunnii is probably the fastest growing tree, followed by Leylandii or Leyland Cypress, the much hated hedging tree. Both are evergreen - Eucalypts will put on around 6 feet of growth in a season, Leyland Cypress about half that. But do bear in mind that there's a price for planting a fast growing tree - they get very tall indeed quite quickly - Eucalyptus around 50 to 60 feet and Leyland cypress even taller if not pruned. As for size when purchasing, don't buy a large Eucalyptus - they grow faster and settle in better if they're not large when you buy and plant.

Western Red Cedar, Thuja plicata would be my choice. Can still look good when trimmed with electric hedge cutter if you want it to look more formal.

Eucalyptus gunni as mentioned is probably the fastest but fast growth can result in it falling over in high winds. For natives Birch and Rowan take about 4 years from seed to reach 8ft and are much more adaptable to our weather.

Paulownia tomentosa is said to be the fastest growing tree in the world, and it has lilac-pink flowers in spring. There is a recent blog about it - suggest you send a PM to the questioner for practical advice.

Paulownia is fast growing - not much between it and Eucalyptus in terms of speed, but Eucalyptus is evergreen, Paulownia isn't, so depends whether you want privacy all year.