Drs generally only pronounce time of death if a resuscitation attempt was made and they’re there to do it. Otherwise I think it’s estimated by ward staff and then an autopsy could clarify. I’m not entirely sure about that bit.

Rest of this is right, but formal confirmation of death has to be done. Normally in hospitals this is done by a doctor who does a very specific examination (check the identify of the patient, check for verbal response, check for response to painful stimuli, check for pupil response, auscultate for heart/breath sounds for at least four minutes while also checking the carotid pulse.

This is what is recorded as the formal time of death, which is why you'll see the ward staff getting understandably anxious about someone who has died in the evening when the on-call hasn't had a chance to come and confirm them yet as noone wants the patient who actually died on Tuesday at 23:30ish to be recorded as having died on Wednesday at 00:30.

Anyone who is competent can do this - including nurses - and each profession has its own set examination to perform which is fairly similar. In hospital however this tends to be the responsibility of the doctors.