If the missed connection is WN's fault - even if it meant there was no other connection that night - I'm confident WN would take care of you. When a delay at one's originating city will likely cause a misconnect, especially a misconnect that causes an overnight at the connecting city, I've seen WN take proactive action and reroute the pax from his/her originating point so that the pax doesn't get stranded at the connecting point.
If it is your fault, the picture gets murkier. If, for example, you got delayed in the airport bar and came up to the gate and reported you missed your connecting flight and there were another flight that same day, my expectation is they would accommodate you on a standby basis. Overnight, or two days later, I think you'd have to buy up to the full one way fare. If the misconnection, especially one causing an overnight, was WN's responsibility, your PNR would be so noted. Absent such notation, they would know it was not their fault you misconnected.
Welcome to FT.