IIRC during the first months of the pandemic most deaths showed a pattern where, from onset of symptoms, the disease would be unpleasant for ~7 days then the patient would seem to recover, and then suddenly worsen around day 10-14. New cases were typically detected around onset of symptoms, so I'd guess that's where we get the general assumption that deaths will trail new cases by 2 weeks.

I'd be curious whether we're detecting noticeably earlier than symptoms now, in countries like Australia with very proactive testing and contact tracing.

Yes most people who get it bad die after about 14-28 days. I don't know the detailed stats, but the average might well be 3 or 4 weeks or so, but the first of them start dying after 14 days. This means when you see an increase in infections the death rate starts to rise after about 14 days, as the leading edge of the rise in deaths starts entering the statistics.

4-6 week delay is what I observed earlier in the pandemic. Once people go onto ventilators they usually take a while to either get off them or die.

If you take the numbers from the UK (here https://coronavirus.data.gov.uk/) and filter to just the 2nd and 3rd waves (testing was poor in 1st wave; lots of people are vaccinated in the 4th) and compute the autocorrelation function between positive tests and deaths, you see a peak around 14 days.