"In the absence of any agreed and contracted process for handling correspondence for "
If you didn't ask for it (specifically, and in detail) you don't get it.
Now I might think that "S**t's accumulating in our offices" would be a flag to contact someone in the DoH and ask them "What do you want done with it?"
Did this happen?
Maybe and the DoH asked "What's it going to cost?" or maybe they were told "We're not sure as the guy who deals with that stuff was let go two months ago."
or maybe they didn't bother.
Remember the Golden Rule of Outsourcing.
"Everything the client forgets has to get done is another opportunity to charge for additional work. "