With Starlink's peer-to-peer capabilities, hitting every single ground station and keeping the satellites from working through new ground stations may actually be quite difficult.
Starlink orbits close enough that they're looking into offering LTE coverage from "space". You don't need a giant dish to access the satellites, which means building new ground stations and reprogramming the network from an unassuming-looking ground device to use them is quite feasible.
The paths of the satellites are rather predictable, though, so your shrapnel attack executed with some precision should clear out enough of them.
The moment you launch a nuke (even if just to set off an EMP), you can expect nukes to come your way in retaliation before your nuke even detonates. Unless whatever war is going on has already gone full nuclear, I don't think nuclear weaponry is a viable move to take out satellites.
