Something else for you to think about, UK.
I rather like the stone used in the dwarf wall. This is entirely personal, but I would just rebuild the wall using the stone again.
It would complement those pavers nicely. It's a lovely rural setting, and I'm not a fan of modern brick. I always call it the "Health Centre" look, especially when the chosen pavers are nicely subdued and rustic.
Rebuilding would mean that I would throw away the level and keep that "random" look that the stone gives you. Trying to level it up would be unnecessary.
But... your question: if you use any modern brick, even rustic, it should always be laid horizontal and level.
The very nature of continuous straight lines in the brick joints (horizontally, and vertically) tend to make it look awkward if they run with the adjacent slope. It would, of course, mean breaking the wall into sections, with one end higher than the other. Nothing wrong with that, but it means having a "stepped" wall along the length of the drive.
A long length of brick wall with sloping horizontal beds would mess with my head.......... but that may be just me ;o)))
Foundation: certainly not necessary for re-building the stone.
Even with brick, there is such little load, that you could get way with what's there already.
BUT............ brick laid level means that you would have to pay quite a lot of attention to getting a level bed to work off. No problem, but bear in mind that it may involve some excavation at one end of each section......... and obviously level at the other end. You don't want foundation concrete rising above ground level... not a good look.
So much to think about, isn't there? :o)))))))