I have suffered from exactly the same situation. Rebooked a flight back to AMS to one day later. Arrived at LHR only to be told I had missed my flight yesterday.

In my case I also did receive a confirmation, and I had also taken that as a confirmation of the change. It even said that in the email. And even though I am very familiar with e-tickets, and I do know that the e-ticket number is supposed to change, I didn't notice. To be quite honest, I don't think the e-ticket number is even mentioned in the confirmation email. I don't believe one ever gets a new e-ticket email when rebooking (Normally only the confirmation is sent if I remember correctly).

Eventually they were able to fix it at the airport, I paid the change fee that I was quoted a few days prior, and all was fine. In my case the new flight was actually still/already added to the booking, just in status 'unpaid'. And KLM failed to notify me in any way that payment failed - online on the website, and the confirmation email both said all was successful.

In the OP case I would not accept this situation. For me it was salvaged, but in the situation described by the OP I would do anything to get that money back. KLM really dropped the ball there and they should at least cover damages.