Depending on your analysis, west can be read as an adverb or as part of an adverbial phrase, defined as a phrase that is functioning as an adverb.
Traditional Answer: West is an adverb modifying the preposition of or the prepositional phrase of Bowes
For example, the Oxford English Dictionary suggests that in the following example (similar to 2 in your question) west is an adverb modifying a preposition ("west, adv., adj, n.1," A.1.b.(a)):
2005 J. Diamond Collapse (2006) vi. 197 Iceland lies in the North Atlantic Ocean about 600 miles west of Norway.
So west would be an adverb that premodifies a preposition. The Oxford Companion to the English Language describes a typical case:
Apart from modifying adjectives and adverbs, some adverbs may modify prepositions (well in He kicked the ball well past the line)
This is the simple answer.
Newer Answer: West is an adverbial that can be combined with other adverbials
It makes sense to treat west as an adverb modifying a preposition or a prepositional phrase. That said, doing so neglects a property of west - it would also function in the absence of "of Bowes." That suggests that west is itself an adverbial of place.
For example, if one took out the measure phrase "a mile" or "about 600 miles," the sentences would remain grammatical:
the accident happened west of Bowes.
Iceland lies in the North Atlantic Ocean west of Norway.
The adverbial phrases indicate direction, and don't need the measure phrase to function.
Furthermore, removing the prepositional phrase would also result in a grammatical usage, albeit one that requires a contextual understanding of the frame of reference - west of what?
the accident happened a mile west. (OK if we're speaking and I can presume "of here," but not good if I'm reading a newspaper)
Iceland lies in the North Atlantic Ocean 600 miles west. (OK if my question was "In relation to Norway, where is Iceland?" but not in more general cases where I might pick Ireland or Greenland as a frame of reference)
So if the prepositional phrase is omitted, there is still a sense that west serves as an adverbial signalling direction in relation to a frame of reference, even if that frame of reference is presumed. In this way, west would be an adverbial signalling direction, and the fuller phrase west of Bowes would be an adverbial phrase.