I see this all the time on traffic from far east to Europe when a Red Sea cable dies, and I’ve seen it from India to Europe too (and seen traffic rerouted via South Africa and up the west coast a fair bit)
Latency is higher, but on the whole things continue to work. Until there’s enough damage - problems tend to be if you cut enough that cables and the routers they are connected to start to bottleneck.
While the number of cables is not large enough to put it out of the reach of many nations, it's also something that no group with the capability of doing it would really want to do: it's a surefire way to invite retribution from basically the rest of the world, while not really achieving much militarily: armies almost invariably have their own communication systems (satellite, microwave, transoceanic fiber whose location is secret, etc).
[1] https://www.channelstv.com/2024/03/16/internet-disruption-su...
But realistically, I would think the US/Americas would be approximately fine. Most, if not practically all, services people on the NA continent use are based in the US from both a corporate and technical perspective. The command+control stuff for distributed systems is probably in the US.
Across the pond(s), yeah, I’d expect more disruption.
20 years later I wonder how many of those are hosted on AWS/GCP/Azure and would break anyway. Probably all but the biggest.