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I understand from this post:

Acheter qqch à/de/pour qqn

that "acheter qqch à qqun" could mean "to buy something for someone", or "to buy something from someone", and that only context clarify the ambiguity.

How do I say the following sentence in French, then, which requires both "for" and "from"?

"Can you buy a birthday cake for my nephew from that fancy bakery we went to last year?"

I would translate the following sentence:

Can you buy a birthday cake for my nephew from that fancy bakery we went to last year?

In french this way:

Pouvez-vous acheter un gâteau d'anniversaire pour mon neveu, à cette
boulangerie de luxe où nous sommes allés l'année dernière?

Q:Can I have more explanations about why did you use "à" in the last sentence ?

A: Yes, I'll try:

à is a french préposition used here in order to describe a destination;

FR: Où allons nous? Nous allons à Paris.
EN: Where are we going? We're going to Paris.

FR: Où allons nous pour acheter le gâteau d'anniversaire? Nous allons à cette
boulangerie de luxe où nous sommes allés l'année dernière.
EN: Where are we going to buy the birthday cake ? We're going to that fancy bakery we went to last year.

The question you linked already tells a way to disambiguate the intended meaning when à is followed by a person:

J'achète un livre à/pour mon ami.

• 
  à : Ceci peut signifier deux choses selon le contexte :
  
  
  
  
  1. ...
  
  
  2. 
     J'achète un livre avec l'intention de le donner à mon ami.
  
  
  
  


• 
  pour : Ceci peut signifier deux choses selon le contexte :
  
  
  
  
  1. 
     J'achète un livre avec l'intention de le donner à mon ami.
  
  
  2. ...
  
  
  
  

However, your English sentence is not a good example if this potential issue as you use a different à:

Can you buy a birthday cake for my nephew from that fancy bakery we went to last year?

Here from doesn't mean from someone but from some place. In such case, there is no ambiguity so you might still use à for both terms:

Est-ce que tu peux acheter un gâteau d'anniversaire à mon neveu à la super pâtisserie où on est allés l'année dernière ?

In any case, the meaning is very often obvious, even without more context than the sentence itself. e.g.

J'ai acheté des chaussures à mon fils (for my son)

J'ai acheté une langouste à mon poissonnier (from my fishmonger)