As far as I can tell, BLAKE2 has effectively all the properties BLAKE3 has (arbitrary output length, keyed hash mode), so the upgrade for communications boils down to negotiating/determining which of the two hash functions to use over the wire (with all the downsides that come with agility of cryptographic primitives); for stored hashes, they have to be recomputed and replaced (or you could store a flag is BLAKE2/is BLAKE3 and update them as you touch hashes, kind of similar to how password hashes are swapped at login time).
Note that BLAKE3 existing doesn't break BLAKE2. It's perfectly fine to just keep trucking BLAKE2, it's just that BLAKE3 has better performance characteristics that make it very attractive.
Though for Wireguard, you'd compete with Blake2b as well, which has the advantage of using 64-bit words. And if you want a fair comparison, you should reduce the rounds of Blake2b down to 8 (instead of 12), as recommended in Aumasson's "Too Much Crypto".
On a 64-bit machine, such a reduced Blake2b would be much faster than Blake3 on inputs greater than 128 bytes and smaller than 4Kib.
The concern about 64-bit machines and using 64-bit word sizes vs 32-bit word sizes really only matters if your 64-bit machine doesn't have SIMD vector extensions. (All amd64 hardware, for example, has at least SSE2.) And as they point out, being 32-bit native really helps on low-end 32-bit machines without SIMD intrinsics.
(Re: the hypothetical, if wireguard were to do a protocol revision and replace Blake2B with this, it would make sense to also replace Chacha20 with Chacha8 or 12 at the same time. I doubt the WG authors will do any such thing any time soon.)