How do these distributed build tools work across the Internet, i.e. outside of a local LAN office setting? Does the increased latency and slower bandwidth become a bottleneck?
They will work over the internet. You should use SSH or a VPN like Tailscale because the native protocol is not encrypted.
Bandwidth and latency effects will have some effect on how well it will scale, but on a reasonably high latency and low RTT connection it's probably still faster than purely local compilation. Sending work over the network takes time but not much CPU, so the local CPU is freed up to do other work.
If you distribute preprocessed code, you need LAN speeds/latency for it to be faster than a local compilation. Otherwise, you need to have the exact same headers in addition to the exact same compiler installed everywhere. It MAY work otherwise, until it breaks in a subtle / obscure way one day and after tracking it down you will think maybe compilation doesn't take THAT long on a single machine :)
So all in all, I wouldn't say works outside of a well-controlled cluster on a LAN.
Assuming you don't use NFS to continuously read and write, I think it would be fine. Compiling is generally speaking CPU bound.
Even the website mentions that it just copies all preprocessed code to the workers, so I expect this to be a one-time overhead per job.
Bandwidth and latency effects will have some effect on how well it will scale, but on a reasonably high latency and low RTT connection it's probably still faster than purely local compilation. Sending work over the network takes time but not much CPU, so the local CPU is freed up to do other work.