To design a blue green deploy how to publish websocket traffic from live to hot swap server
Websockets don't really have any handling for this per se. You could build your own system but generally the idea for HA is that if anything happens you'll set the client up to automatically reconnect. There will be some non-zero downtime during a blue-green swap but if your app is well built it should handle that already.

If you wanted something more smooth, you could build a message in to your protocol to connect to a new server and then drop the original connection.