Last report I saw was PP are taking 6-8 weeks.
Emergency "travel document" is just that , for Travel, and is issued by the Embassy usually to allow a citizen to get home, not for Visa application purposes.
When you get your NEW passport, you will get an accompanying letter to give Immigration.
This will enable smooth transfer of permisson to stay etc. to the NEW passport.
Also be sure you change the passport number that may be held at any Bank where the passport is used as ID.