I try to read as much as I can on on the internet and I often see 読み込み中 or sometimes 読み込んでいます to mean reading/loading data when websites or programs are loading things.

This got me thinking, are there any differences between the two and when would you use either? I have just took them both to mean "currently doing x" and I take them to mean the exact same thing when I read it.

Whenever I want to say something like I'm currently drinking milk I would always think of: ミルクを飲んでいる。 and never something like ミルクを飲み中だ。

中 is only used in more official-y sounding things. The only time I would use 中 for drinking milk would be to be overtly formal in an effort to be funny.

Yeah, ～中 is used mostly in formal/official (= 硬い) contexts.

お取り込み中失礼ですが・・・ "sorry to interrupt what you're doing, but..."

In Japanese, that's the tendency in general with kanji.
More on-yomi readings = more 硬さ

My husband says 仕事中、レッスン中 etc. to me all the time so it's definitely not just for formal sounding things. I just think 中 is usually used with kanji compounds and not verbs most of the time, i.e. 仕事中/仕事している but yeah, I think attaching it to the end of a verb does make it formal.

Do you know the differences between kango and Yamato kotoba?

ている is Yamato kotoba. Meaning it derives from the native Japanese language that existed before Chinese influence. 中 with the reading ちゅう is kango which is all the Japanese that came from Chinese.

***in general*** kango is considered more formal than Yamato kotoba. You'll see kango used with higher frequency in written Japanese while the usage of Yamato kotoba will occur more often in spoken Japanese.

Here's an example of using お茶漬け中 to comedic effect.

I don't think there's really a formal rule regarding the usage and there isn't really a politeness level associated with it. You can say 作業中でございます and it'd be perfectly polite, or 作業中だよ to be more friendly. ミルクを飲み中だ sounds very unnatural, though.
It seems like 中 sounds best after a complete phrase or 熟語. I'd say if you couldn't answer the question "何してるの？" with <word> alone, don't use 中 on it.

For example,
Q: 何してるの？
× A. ミルクを飲み
○ B. ミルクを飲用

So ミルクを飲用中 is fine.

And of course, ミルクを飲んでいる中 would be redundant and sound weird.