We know so far Nvidia will set aside 1GB of VRAM for it.
Where do you get this number from?
I would assume the BVH can have any size depending on scene, also larger than 1GB? Likely 1GB is some kind of average from current games?
Beside BVH, it would be interesting to know if they use a stack per ray (likely) and where they store it.
The only option i see is L1 (which they also use for LDS / texture cache), likely reserving a fixed, small size per ray and if this becomes too small the traversal algorithm needs some restart fallback to still give correct results.
I guess this will be very similar for AMD, maybe the TMU patent did even mention LDS. (They also added some short stack traversal variants to Radeon Rays.)
Also interesting: Does NV use binary tree or MBVH?
AMDs TMU patent mentioned a branching factor of 4, so the tree would have much less levels than a binary one, and memory access can be made more coherent.
(Still, i wonder why not 8, because that's a nice number to divide 3D space like an octree, while 4 would be nice for 2D like quadtree - must have technical reasons.)
However, it's just details. I do not expect a big difference between NV / AMD or desktop / console from the programmers perspective.
Only exception is AMDs option for programmable traversal to enable things like stochastic LOD, fallback to reflection probes if ray takes too long, things like that.