I would recommend that under the weed-suppressing fabric you spread a very thick layer of mulch such as spoiled hay (although that does contain weed seeds), or straw, or strawy manure. This will keep the light out more effectively than weed fabric alone, and will rot down to make a nice friable fertile soil to plant your trees in.
Whether the weed fabric can fully suppress your weeds will depend partly on the type of weeds you have. Creeping thistle, nettles and couch grass will simply come in from the edges and recolonise, and couch in particular will grow on top of the fabric as well as under it. Make your covered area as wide as you can. Even just a year of weedfree growth will help your trees to establish, although they would probably survive without it. We planted so many trees and hedging plants that we simply couldn't afford to cover the ground under them, and like you we don't use weedkillers, or any other chemicals on our land. The trees and hedges we planted are well established now and shade out the plants below to some extent, but some grass now they are big also provides cover for wildlife - partridges and so on.
Bank voles in particular love to live under weed-suppressing fabric, which gives them an ideal place to shelter while they gnaw their way through the bark of your newly planted trees, so be sure to use spirals and get them well down the trunks, protecting the parts under the fabric.