Timescales for ESA decision

I have been supporting a young man through his ESA application. We had the face to face on 6 March. As we hadn't heard anything I rang DWP today. They said that they hadn't yet had the 'paperwork' from the assessment centre and that it could take up to 4 weeks for it to get through to them. I am surprised at this as I thought it was all on line and as soon as the face to face was done then the 'paperwork' was available to the decision maker. Certainly that's how it seemed to work with his PIP application.
Sounds like we still have a few weeks to wait for a decision....
Did I understand what I was told correctly?

Thank you for the reply. As suggested I rang maximus. They assured me that the file had been sent on and they no longer have it. I then rang dwp again who said they hadn't receIved anything.

Does this sound normal?

I'll give it a week and ring again?

Any comments or anyone else has the same experience? . Should I worry?

Well that was interesting. Despite both the assessment centre and dwp saying they didn't have the 'paperwork', and me fearing the worst, all of a sudden money turned up in the bank and today we have had the successful decision letter. This has been a long and protracted affair (the claim took a year) but thanks to the support of this website, we have come through. Thank you, we are very relieved.