Next does run a node server, whether it’s hosted serverlessly or not. And assuming it supports web sockets you’ll just hold a connection open for the duration
There's no such thing as the cloud, just someone elses computer There's also no such thing as serverless, just someone's cloud that they use to share a pool of servers among lots of people
Afraid I can't help much on the hosting front though, particular in Nextjs. It's a lot of layers of abstraction so i would start simple (get it working locally for instance)