For Starlink, maybe. However, I'm sure nuclear submarines operating in the arctic (and the US Navy in general) as well as forward operating US bases, would love and be willing to pay for Starshield to cover everywhere.
Pretty sure they don't have any comms signals and wouldn't connect to that unless absolutely necessary emergent situation. I thought they operate in the dark.
Subs do pop up to make status reports or check-ins. They do operate independently in that command knows what area they patrolling but not exact locations at any given time.
Subs (at least the US ones) also use "strategic" LPI/LPD milsat comms instead of commercial satcom. You don't want your enemy to geolocate your sub fleet whenever they phone home.