How do I recover from accidentally missing a credit card which made my account delinquent?

Why did you submit a dispute to the credit bureaus? Do you believe that the reported information is incorrect? It doesn't sound like it from your description. It sounds like the credit bureaus are correctly reflecting the fact that you made a late payment. You would send a dispute to the credit bureaus if you believe they are reflecting incorrect information-- an account that isn't yours or a late payment that was actually on time.

Have you called up the credit card company? Normally, if you've got a solid history of payments, call them up and explain the situation, and ask nicely, they're more than happy to remove a late payment from their system (which will then get sent to the credit bureaus the next time they report) and to waive any late fees. Particularly if they should have taken the payment from the new account (I'm not completely clear on the timing here-- did you update the payment information really close to the payment date?).

The credit card company has an interest in keeping you happy since they make money when you use their card. The credit bureaus derive little benefit from keeping you happy-- you're the product for them, not the customer.