How did you go about filing a claim? How did you find out it was lost (in post or in the DVLA office?).
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I called them and told them that I have not yet received my document so they said they will make a formal enquiry with the Postal Service, which took another 3 weeks or so. They then wrote me a letter saying that they believe it has been lost in the post, and the letter did say that they will reimburse the reasonable cost of replacing it.
For me, the reasonable cost included the train fares to the US embassy in London to get a new passport application done, the cost of getting the new passport posted, and the fare to Croydon to do all that biometric stuff again.
It was such a tense 3 or so months for me. My US passport was brand new--less than a year old when I lost it.
Oh, you need to go and make a police report as soon as you get the letter from DVLA stating that the passport is really missing.
PM me if you have any other questions.