Try the following link on desktop (mobile seems to botch it) after removing the spaces towards the beginning and possibly replacing the letters ‘nz’ with your destination country’s two-letter shorthand:
https :// mydhl.express.dhl/nz/en/tracking.html#/track-by-reference
under ‘Shipment Reference’, enter your PEX reference number without the PEX part and without spaces (just the numbers), and finally under “Date Range” you can enter starting date as the date of application and end date the current date.
If the above link doesn’t work, Google HM passport DHL. You should reach an FAQ page that contains a link that redirects to the tracking page.
Had an application in Durham a few months ago and it took about 7 days after printing before it showed in tracking on DHL’s.
Also, I probably had an issue with the tracking page not properly refreshing. Best to revisit the page and reenter the details when you wish to check again if DHL has your passport.
You’ll be able to see tracking for both shipments, the new passport, as well as the supporting documents which follows later (one day after in my case), using the same PEX reference number