Daisy Mae has the best answer for anyone who wants instant effective screening. if you draw an imaginary line at 45 degrees from next door's upstairs window to points in your garden, you find that a well placed pergola with trellis or bamboo cane 'roof' gives you more privacy than a trellis fence within the garden.
If you have time to wait for a standard tree to fill out its crown a spreading tree placed sort of in mid-garden is a shield against upstairs windows. Both the above options make the garden more shaded in parts.
To increase the privacy of the boundary fence I believe a row of 12 foot high pleached trees such as evergreen holm oak is perfectly within planning regs if the trees are well spaced. Quite expensive depending on number of trees and paying the gardener to dig the holes.
Finally, it's within planning regs to erect a gazebo, summerhouse or shed next to the boundary fence. This can be the cheapest option to give max privacy if you can't afford a well made pergola.
My son and I were discussing the very topic yesterday evening!