It will take a long time and a huge number of human work years to achieve once we decide to do this.
Yet anyone who looks at the globe knows that the present well watered ecosystems are a small fraction of the globe’s land area. The Sahara leaps out, but so too does Western North America and most of Asia. Well watered is not how you would describe most of the Globe. Yet our objective is to make all of it well watered.
This will have a major impact on global climate and most pronounced in the Northern Hemisphere. Most importantly, all this woodland will be absorbing energy into its biology. This is sunlight that has previously been reflected back out into space and lost or at best used to warm the atmosphere.
Thus we can expect the climate zones to migrate north and a huge moderating effect to be observed. I would expect a full spring breakup and flushing to develop in the Arctic of all sea ice every year. We are close to those conditions now and a flushing of excess heat would do it quite nicely. We saw it in action in 2007.
Such a transformation would certainly moderate the northern climate and create European weather to the higher latitudes in North America and Russia where we are already used to dealing with shortened growing seasons.
We already have surmised that a covered Sahara during the Bronze Age gave us the climate optimum of the time that lasted just to the end of the Bronze Age. That acts as a good check on our expectations but also tells us the correct direction of change.
In practice woodlands and general growing environments that are not arid bring their own natural microclimate. Properly managed, these can cover most of the globe.