Just having a look at my mapping software for you.
OK, will only comment on areas that I know so that means from Warminster onwards you’ll be using the ridge track which is mainly stone-based. There’s a good few miles of this and it’s easy pedalling. The grassy ridge at Golden Ball Hill should be OK.
There’s a bit now that I don’t know but Easton Hill will probably be mucky due to farms. From here onwards you’ll be fine, ie through Collingbourne Kingston. However when you reach Inkpen Hill, this is full of ruts but they will disappear.
You’ll effectively be following Wayfarer’s Walk and this consists of much grass. Again there will be some muck due to farm use. This is a fantastic area, really gorgeous.
The Candovers are currently very overgrown with the odd tree down blocking a trail or two but there’s always a way. From here to Alresford will be fine, again the odd tree down.
Edit: agree with comment re Wiltshire trails, however Hampshire trails are about the worst I’ve seen for donkeys years.