Lookingforadvice123 I had a similar email to you, saying that my application had been received 5 June (by Belfast) and was put before the examiner on 6 June. When I called Belfast yesterday (23 June) they hadn't even looked at my application yet. I would believe what you hear from the Passport Office dealing with your application.
By the way, I had a response to my on-line query about when they calculate the 3 week turnaround from. Time runs from the date that the Passport Office dealing with your application receives it, not from the date you submit it. Seems pretty unfair to those of you who have long gaps between sending the applications off and it actually being recorded as reaching the Passport Office.
So now my 3 weeks are up on 26 June. Planning to call and chase again on that day.
I thought that you could request fast track earlier than 7 days before travel if you are willing to pay as long as it was practicable (e.g. they weren't waiting for documents from you, checking a countersignature etc)