It's a sliding scale between immediacy and accuracy, with a very high upper length of time for perfect accuracy. I work with death record statistics for a state government, and our reports use data from 2 years ago. A big reason is gathering reports for residents who died in other states.

I know the CDC has a good system for sharing infectious disease diagnoses (https://wwwn.cdc.gov/nndss). I don't know if there's a similar rapid-collection system for death records. We usually don't need to track deaths with this urgency.

A possible compromise is to model a delayed reporting factor. Not sure if there could be a good enough model, though.