Hi:) Having your explanation at hand I would suggest "over-worried".

Or just 'regret'.
To tell the truth, I just doubt the appropriateness of the original term because I have never seen a person using the term 残心 like that and also because I barely found few examples using the same expression on the internet.
I'd just use '心残り' if I were you, but the translation wouldn't change even in that case anyhow.

To scale the level of involvement of a nurse to her/his patients, the nurse's state of mind is discussed and gauzed. Questions are: "Do you still worry about the patient?" "Do you regret that you should have done something differently for them?" "Do you have a lingering issue to concern about the patient?"

To have 残心感 can be expressed as "to have some lingering reflective/regrettable issues."