There are numerous fast growing trees available to buy, but there are additional factors to consider.
If you are looking for the fastest growing trees, Paulownia tomentosa and Eucalyptus gunnii are probably the fastest and you can expect in excess of a metre a year. If you looking to create a screen, Populus nigra Italica and Cupressus leylandii are among the fastest growing hedging plants, with the former producing mammoth columns and the latter a dense screen.
More traditional native trees such as Betula pendula, Salix x sepulcralis and Sorbus aucuparia grow quickly and provide good value for wildlife. Then you get the big American beasts, the giant Acer rubrum, Liquidambar, Nyssa and Liriodendron with their amazing autumnal displays.
It's no good buying a fast growing variety and then slowing growth through improper care, so ensure you remove competing plants from the radius of the rootball, which increases in distance every year as the tree's root system spreads, and laying down mulch to trap heat. Removing fruits in a tree's early years will force it to put more energy into vegetative growth.