The fact that it is a 2-of-2 multisig is relevant only because the UTXO - that is actually the channel - must be spendable only if both parties agree on that or some other conditions are verified (like in force closure). The onchain broadcasting actually happens automatically once you or the counterparty decide to close the channel, which can happen whenever you need to (that is the game theoretical assumption behind the security model of LN and the reason why it is actually a self sovreign protocol).