As folks have pointed out here, there are a lot of variables. Are you envisioning this... Okay, let's start at the beginning, because I'm guessing you don't know anything about Chinese (or Asian in general) family dynamics from these questions. I'm half Chinese (from me Mum) and in no way can I speak for all Chinese, Asian Americans, or any other group.
Firstly, what on earth do you mean by address? Are we speaking in English or not? (Because you can't technically speak Chinese, you're speaking Cantonese, or Mandarin, or Shanghainese, or Taishanese etc etc) It's not uncommon for ABC's (a term for Australian or American Born Chinese) to speak, or not speak, their parents mother tongue. You also get fun dynamics like parents speaking in Cantonese and children responding in English. One of my full Chinese cousins would do that (and now super regrets it because he needs to use Catonese in his life.) Quite a few of these dynamics have to do with white people racism, which is a whole other can of worms, and this post is already going to be long.
Now for me, all of my mum's close Chinese friends are auntie or uncle, in chinese. This term of course varies dialect to dialect, and also culturally. For example, in Cantonese I'd call my aunties "ze ze," when speaking directly to them, and "[name] ze ze," when talking about them. (Note one, romanization is stupid, this is not pronounced how it looks, unless you're a witch.) Ze ze is technically older sister, but it's polite to call your aunties that. Again, I can't speak for other dialects and Chinese folk, but Cantonese (at least HK Canto) is just like that. It's also totally acceptable to try to get your waiter's attention by saying, "Pretty lady," or "Pretty man." (Oh, also Chinese is a tonal language. Dialects vary on how tonal, but Cantonese is very tonal. The word for pretty and thug are the same word with a different tone, and guess who's made that mistake?)
Now for anything in English? Just auntie or uncle. I will note that I find us Chinese folks elevate folks to auntie or uncle faster than Americans seem to. All my parents friends were auntie/uncle [first name]. If they're not close enough to be an Auntie or Uncle, why hang out at all??? My yougner cousins, when speaking in english, will call me, "Nat cousin." It's cute. It's also polite. (Leaving off titles is impolite, as well. Calling an auntie Ms. Name is really rude. Always auntie.)
So, basically, I can't answer a lot of the "what would you call people," without knowing if we're speaking English or Cantonese. BUT Chinese does have very clear words for a lot of family relations (like uncle on your mother's side is a different word than uncle on your father's side). I'll give it a try though!
When mother and daughter are talking privately, how would mother refer to her friend? (e.g. by first name, or first and last name, or nickname, or title...?
The mother would say, "You know your auntie [name.]" I've literally never heard my mother say, "My friend," and it's always your auntie or just auntie. Huh. Hadn't really thought that was super standard english till it's pointed out.
What would the daughter call her mother's friend in this private conversation with her mother? (e.g. some version of 'auntie' or...?)
Also just auntie.
How would the mother address her friend directly (e.g. first name, some form of diminutive nickname...?)
Oh they'd have nick names. These are realllllly specific. My mum's little fish. Her youngest brother has a nickname, but is usually called dumb shit instead by his siblings. My mother calls my sister and I both unsavory nicknames, but none of the rest of our family can use them. I used to have a friend who called me fat girl and I'd call him fat boy. So, yeah, varies a lot.
How would the daughter address her mother's friend directly?
AUNTIE.
How would the mother refer to her friend's nephew when talking about him to her daughter?
So the mum might bring it up, "Do you remember Auntie [name]'s nephew, [nephew's name.]" Which brings us to...
How would the mother address said nephew?
A fun dynamic I haven't mentioned yet! Generational hierarchy. You're always polite to old people. You just are. If someone is older than you, you be politer. So the mum can call friend's nephew just by his name, but he needs to call her auntie.
What would the mother call her daughter when speaking with the daughter directly (first name or some other term?)Again, nicknames. Sometimes your name. It's like in English too (although this could be something my mum picked up from living in the west), where when you're being yelled/it's serious at it's your name not your nickname.
How would the daughter address her mother when speaking with her directly?
Mum. In either language.
Are there any particular turns of speech or phrases that might distinguish the mother's speech, make it sound more authentic?
Oph, so yes, and no. There's certainly common mistakes in English the Mum would make, but unless you're Asian I wouldn't recommend this. Unless you do a lot of research and actually talk to some more Chinese folks at best you'll come off racist. At worst you'll come off really racist.
There are idioms/phrases that are Chinese specific that my mother would say in English translated poorly. One was, "Your head." It's like... "That's ridiculous/you're crazy." So a conversation might go like this.
"Can I go to Sam's house?"
"Your head you're going to Sam's house." Which of course means, "You're crazy if you think you're going to Sam's house." The "your head" can also go at the end of the sentence. It's flexible!
She'd also say, "Frog in a well," in English which is an idiom about how a frog in a well only knows the well, so it doesn't know or understand the world outside. It can be used to call someone narrow minded, but she'd also use it in place of, "Can't see the forest for the trees." I imagine every family has their own ones of these.
Later, the daughter gets together with a colleague who is not Chinese-American (Caucasian). Would the mother object to having a white son-in-law? Is that even an issue these days in Chinese-American families?
Yeppers and it varies a lot. Again, huge can of worms if you want me to go into racism issues, but my mother is a bit cheesed that my sister and I are both dating white people. She did tell us not to marry first born Chinese sons as well (this is a somewhat my mother specific gripe, to do with how first born sons are raised differently in some families, very spoilt and ill mannered, and not a good match for women who value independence.) But my mother has bluntly told me she has more issues with me dating a white person than my partner being a woman.
I have friends who were disowned for dating outside their race, and I have friends whose parents couldn't give a shit. I even have friends whose parents encouraged them to date white people, since white people run the country.
Hopefully some of that is helpful! Again, I really want to stress that this stuff can vary, a lot. China, like anywhere, is not a monolith. There's lots of different cultures, and different and valid ways to be Chinese. Please do try to talk to other Chinese Americans (and there's other research you can do like reading forum posts or reading books by Chinese Americans.) But really, the biggest take away is how we're all different and research is good!