Many people overseas have to have a valid passport as a condition of their residence in their second country, so I thought it would be handy to post this to give people an idea of turnaround times, especially as the passport office will advise you to come back at a later date due to COVID-19 if you are not travelling during the summer.
The passport renewal website is pretty easy to use and states what you need quite clearly. I would advise going to a photo booth that allows you to save the passport photo online as taking a good photo with a mobile phone is pretty difficult. The cost of the passport is £86 plus a £19.86 courier fee, as well as whatever it costs you to mail your passport to the UK office (I spent NT$390 or roughly £10 for signed delivery of my old passport and a colour-photocopy of my other nationality passport).
Passport renewal website: https://www.gov.uk/apply-renew-passport
July 18, 2020: I filled in the online form with a digital photo.
July 19: Email prompting me to send my documents.
July 20, 2020: I mailed my old passport and a colour photocopy of my second nationality passport.
17:00 July 28: Parcelforce stated that the parcel arrived at the Liverpool HMPO.
July 29: Email prompting me to send my documents and notice informing me that due to COVID-19 I should receive my passport in 8 weeks.
August 2: Email prompting me to send my documents.
August 14: The office acknowledged receipt of my documents.
August 16: Notice of application being processed.
October 19: Application approved and passport ready for printing.
(9 weeks and 1 day from notice of application being processed; 11 weeks and 6 days since Parcelforce delivered the parcel to Liverpool HMPO).
October 21: Notification that passport has been printed and sent.
October 26: Received old voided passport in the mail.
October 30: Received new passport in the mail.
That’s 3 months and and 11 days (104 days) from filling in the online form and posting the documents to receiving the passport, and 3 months and 2 days (94 days) from when ParcelForce stated they delivered my documents to receiving my passport and 2 months and 16 days (77 days) from acknowledgement of receipt of documents to receiving my passport.