James Farrar wrote:

There is a huge variation around the country in the local authorities'
requirements for minicabs. I have a friend who use to run a minicab
business in Aylesbury, but now runs a similar business in Middlesex.

Time traveller, is he?

(for m.t.u-t'ers, Middlesex hasn't existed for 44 years)

It exists. The Local Government Act abolished only its council.

So where is the Lord Lieutenant of Middlesex?

Or is Derby not in Derbyshire?

The Lord Lieutenant of Derbyshire's responsibility includes Derby.