Tom had had a car accident the week before and he had been ill since then.
This sentence tells us that straight after a car accident Tom became ill and that he was still ill about seven days later.
We know the sequence of these events. We do not know when they occurred (last month, last year, last century). In the absence of further contextual information, therefore, neither last week nor a week ago can be used. Instead -
Tom: "I had a car accident and I have been ill for about a week".