How can I have the first task in a flow specify a timestamp that subsequent tasks can access? Basically I want all tasks to be able to read/write data to the same directory, with the directory have a name like 2022_06_09__113826 based on the approximate time the flow started.

You want the task to return the timestamp, and then simply pass it into subsequent tasks as a parameter

Tim is right, or you can manipulate something like

prefect.context.scheduled_start_time