What to do with/to this driveway?

Hi ya Tok,
Welcome to the forum, its always a bit of a toss up in these situations but from your photos I would take up the old stuff down to about 250mm from Final finished level. By the looks of what you have now that means you probably will be digging through the original subbase and some soil too.
Give yourself 150mm of compacted thickness roadbase and 100mm thick 32 mpa concrete with steel mesh. You could use steel dowels at the joints if you have a root problem.
If when you are excavating you come across a large root, >150mm diameter I would think twice about cutting it as it may weaken the tree instead use that as your subgrade level. Try to get 98% compaction of your roadbase, it may be worth a hire of a small ride on vibrating roller if not compact 50mm layers with a flat plate compactor using plenty of water, Recycled roadbase is cheaper ask for equivilent of DGB 20. Dont forget your control joints.
What I've just dribbled on about is a bit of an over kill but my moto is IF ITS WORTH DOING........

Resurfacing over the top would only be a temporary patch, so long as there are large tree roots underneath near the surface. [shrug]

I tend to agree with BThumb, but then you run the risk of damaging/destabilising the trees...

Mebbe a compromise? Perhaps some bedding sand compacted over the top and paved with individual bricks/pavers/slabs of slate? It'd probably be a bit cheaper than stripping/replacing although more expensive than just resurfacing, but IMHO it'd be more appealing to the eye and, most important of all, it's not a permanent structure. A few years down the line, it'd be a simple matter to patch any future damage or rip it all up and use the materials elsewhere.

(Personally I'd go for natural edged slate in mixed colours. That way, any dips/rises/pot-holes just add to the natural look. The straight edges of bricks & most pavers, on the other hand, tend to exaggerate any discrepancies and look 'orrible. Well... that's my opinion, anyway. )