It appears your flight is boarding, so you should be OK.
Since United endorsed your ticket to Delta, I would continue to work with DL. In similar situations where a flight may be cancelled, I will go to the web site and see what flights are still available. This way, if the flight is cancelled, or continues to be delayed, I can offer options to the agent. Generally speaking, they will try to accommodate you, especially if you have status with the airline.
For example, in your situation, if you go the Delta's web site and plug in MSP to NRT for today. There are about 9 flights out of MSP (all 1 stop connections) that would get you to NRT. Since you have 4 J tickets, the only flight that may have 4 J tickets available is the LAX-NRT flight later tonight, DL 2321 (MSP-LAX) connecting with DL 283 (LAX-NRT). This would be my backup Plan.
If the MSP-NRT flight cancelled, I would immediately get on the phone to try to rebook on the backup plan.