Existing Lightning channels are definitely a problem since they’ll exist on both chains and therefore could get into a bad state. In the case of Lightning, the winning move is probably to close channels quickly, split, then re-open on each chain.
Discussion
OK start educating all those thousands of public node runners and also other thousands of private node runners to coordinate them all closing their channels in the same time and in a cooperative mode.
Nobody is compelled to do anything. Two nodes sharing a channel will continue to operate just fine. The only issue is that Lightning payments will effectively be auto-replayed on both networks (in a sense) until the Lightning network bifurcates.
It’ll be messy, but it’ll work. People who are most concerned with replay will split and reopen fastest.