1. Someday, I will run an Italian restaurant near the beach in London.

There are two prepositional phrases in this sentence. One is 'near the beach'. The other is 'in London'. Is 'near the beach' an adjective phrase or an adverbial phrase? Is 'in London' an adjective phrase or an adverbial phrase? Would you explain them in detail? Do they have structural ambiguity?

Yes, the grammar of that sentence is ambiguous, though the meaning is not.

However, if you replace London by some city or town that is known to have multiple beaches, then the presence of the definite article in front of beach (the beach) makes the adverbial reading more likely: the alternative would tend to imply that there is only one beach in that locality.

In Malibu is a preposition phrase (PP). Although a PP can function as an adverbial, it can never be an adjective phrase, since the head of an adjective phrase must be an adjective. (In the terminology used here, a phrase whose head is an adverb would be called an adverb phrase, see e.g. here for more.)

A simpler sentence

Let's consider the simpler sentence

[1] I run an Italian restaurant near the beach in Malibu.

(As in, Malibu, California.)

As I said, we can read the PP in Malibu in two different ways.

The reading in which it is an adverbial

In one reading, the PP in Malibu is an adverbial. This reading is the only possible one if this PP is fronted:

[2] In Malibu, I run an Italian restaurant near the beach.

The reading in which it is a post-head modifier in an NP

In another reading, we interpret the whole word group

[3] an Italian restaurant near the beach in Malibu

as a noun phrase (NP).

I don't know how to force that reading while keeping in Malibu as a PP. The closest I can come up with is

[4] I run an Italian restaurant near the beach that is located in Malibu.

^{One tricky issue, which [4] makes manifest, is the definite article. Malibu has many beaches; indeed, I have been unable to think of a city that has a beach, but only a single one; normally it's either none or multiple ones. This is not a problem in the adverbial interpretation, because we routinely say the beach even when we don't really have a particular one in mind (A: Let's go to the beach! B: Which one? A: I don't know, either the one we went to last week, or the one a bit more to the north.) But when you say the beach in Malibu, or, even more clearly, the beach that is located in Malibu, you are implying that there is only one such beach. So you should either change the beach to a beach, or supply additional specifying information, such as the beach that is located at the southern edge of Malibu (alternatively, the broader context could provide such additional information; maybe the reader already knows which beach in Malibu you mean). I will ignore this issue in what follows, but it seems to me that this could make the adverbial reading more likely if the locality has multiple beaches, like Malibu does.}

In [3], the head is restaurant, and it has the following PP as a post-head modifier:

[5] near the beach in Malibu

That PP, in turn, consists of the head, the preposition near (that word can also be an adjective, but here it's a preposition), and the complement which is the NP

[6] the beach in Malibu

The NP [6] has the head beach and a post-head modifier, namely the PP in Malibu.

The only adjective phrase in [1] is Italian.