When I lost my passport I couldn't get an emergency one immediately. I lost it on a Sunday. It was a local bank holiday on the Monday. First appointment at the Embassy was available Tuesday. I needed photo ID for that appointment otherwise I had to get some other proof of identity to even allow me into the Embassy. I was bloody grateful I had my driving licence at that point due to how long getting the relevant proof would take.
Then my application for a replacement passport (rather than a new first time passport which always takes longer as you need to verify identity with photos and signatories) was quicker because I had a photocopy of my original passport. Otherwise I'd have had to have prove my identity with other documentation and the application for an emergency passport would have taken longer.
As it was I got it done within 24hrs of the Embassy appointment. But it shocked me the procedure if I hadn't had all the documents I needed. You'd need to get to a location with an Embassy (at your own expense), book an appointment with the Embassy online (and hope they had a free slot in the near future), then prove with photo ID or equivalent who you were to get into the Embassy and then have the right documents with you to prove your passport and identity were definitely you before they will issue you an emergency passport.
So even with everything in my favour it took time. And I had to pay for extra nights and new flights and miss work...
If you didn't have ANY paperwork and were outside the system in acquiring paperwork it would easily take ages. Because THAT'S normal.
The British Embassy will not pay for your transport or accommodation if you have issues with your passport. That's your problem in the majority of cases. It's exceptional if they give you any kind of funds. And that because of how many chancers try to take the piss.
And as difficult as it was, even in ideal circumstances, I do feel this is the right approach simply because otherwise it would cost a fortune to bail out the feckless and piss takers. The service is there as a basis minimum but it has to prevent and protect British citizens from economic fraud and identity fraud. The importance of PROVING you are who you say you are shouldn't be underestimated.
It's exceptionally stressful when you are caught in a mess and don't know if your passport has been stolen. You WANT to know that no one else but you can travel or do anything else using your identity. Your passport is the potential gateway to someone fucking you over at a later date. The system HAS to be robust.