When she got into the car at five, ready for the fifteen-mile drive across London, it wouldn't start.
What does this stretch of language represent? Is it an adverbial?
Sample sentence: When she got into the car at five, ready for the fifteen-mile drive across London, it wouldn't start.
The phrase describes what state or condition she was in. It is not a clause.
It functions like an appositive adjective, but is phrasal.
An Appositive Adjective is a traditional grammatical term for an adjective (or a series of adjectives) that follows a noun and, like a nonrestrictive appositive, is set off by commas or dashes.
Since OP mentions the fact that the phrase does not "follow...directly" the pronoun it appears to modify, consider the following:
Before she took a bite of the shoe, hungry as can be, she said grace.
Do not imagine yourself to be a naive parsing algorithm. Clearly any sane human being would know that shoes do not experience hunger (at least in this universe), and would understand that hungry as can be must modify she, not the shoe.
When she got into the car at five, ready for the fifteen-mile drive across London, it wouldn't start.
The element in bold is a predicative adjunct: predicative because it refers to a predicand ("she") and an adjunct because it's an optional modifier in clause structure, i.e. it's part of the VP, not the subject NP "she".
I think it might be misleading to call it 'adverbial', though 'adjunct' is really just a more appropriate term for many of the items that traditional grammar calls 'adverbial'.