Typically psychiatry is not involved in the work up of delirium (but we should be and by necessity may need to be if the patient is misdiagnosed or mismanaged). It would be very unusual for psychiatry to be consulted for workup and management of delirium explicitly. In fact it almost never happens unless its a really bad hospital. Instead, psychiatry is consulted for another reason (e.g. "rule/out late onset schizophrenia", "management of agitation", "decision making capacity", "patient trying to leave AMA", "suicidal ideation", "flat affect") and in the course of the evaluation, we will recognize that the underlying syndrome is delirium and make recommendations for management of associated symptoms (e.g. agitation, hallucinations, delusions) and will recommend general behavioral recommendations and further workup.
Imaging is not typically part of the evaluation of delirium, with some notable exceptions. But we certainly don't (and shouldn't) scan every confused geriatric patients. In younger patients with altered mental status imaging may be useful.
In some institutions neurology is very involved in delirious patients, but this is the exception rather than the rule.
For complex cases input from multiple specialties including neuro, psych, ID, rheum, nephrology is essential.
Psychiatry can also help when there is a "psychiatric" etiology to the presentation, e.g. serotonin toxicity, neuroleptic malignant syndrome, Ganser syndrome, factitious disorder, hysteria, hystero-psychosis, factitious disorder, malingering, catatonia, delirium-catatonia, acute polymorphic psychotic disorder, boufée délirante and so on.