I have a hunch that the InEng (IE/IndE) usage is derived from the polite valediction, Best wishes, i.e. “I hope everything goes well for you”, and the phrase “send her/him my best wishes”. In both expressions two objects are not required, it is understood that the speaker is saying ‘I would like to send you the best of luck today.’
The InEng usage of ‘wishes’(noun) sounds almost Italian to my ears. In Italian the expression is auguri, and it's very common to just say to one another Auguri! when it is Christmas or the New Year. Tanti auguri is literally ‘many wishes’ which Italians say to each other on birthdays, it is also used to say ‘congratulations’ and ‘good luck’.
But do BrEng speakers use this form? No, not really. Before joining El&U two years ago, I had never heard this form of exchange before; e.g. “Did you wish mum?” In none of the English coursebooks or grammar books I have bought over the years, have I come across this usage, so my guess would be that it is currently considered non-standard English. But whether native speakers would understand it, is a different question, and I'd say ‘absolutely!’
In "I wish my colleague", there's nothing to decypher. Context will tell us if, for example, the colleague is retiring or on maternity leave.
The form has also entered the dictionary. In Cambridge Dictionaries Online
wish verb (GREET)
[Transitive] Indian English to welcome someone with particular words or a particular action:
He wishes me every morning.
And its usage has been recorded by linguists and experts
Register Variation in Indian English
By Chandrika Balasubramanian
Indeed, Nihilani et al identify 1000 items (many lexical) in their lexicon of usage which they claim are “are used in a distinctive manner by large numbers of educated Indian speakers of English”
Wish: “IVE (Indian Vernacular English) speakers often use this verb where a BS speaker would say ‘greet’. In BS (British Standard), what is wished must be expressed...”
- He wished me when we met this morning.
- He's stopped wishing me
How might a native speaker say the following sentence in British English?
Ever since we broke up she doesn't even wish me on birthdays.
- Since we've broken up, she doesn't even call me on my birthday
- She never calls. She doesn't even wish me ‘Happy Birthday’ since we split up
It would sound natural to say "That's not gonna happen" when your friend says "I don't want you to turn into a stranger."
"That's not gonna happen" is known as a "colloquialism". It's an informal phrase that would almost never be interpreted literally. It is, however, something that does imply, as you've stated, that you will do your best to ensure it doesn't occur.
Best Answer
All three of your examples use a "that" which is a marker of clausal subordination: in your case, that "that" marks the beginning of a declarative content clause.
Sometimes that "that" marker is obligatory, sometimes optional, sometimes not allowed. There are some related rules on this, but there is no one simple general rule. It's something that native English speakers just pick up while speaking and listening and reading.
Note: Your "that" is NOT a relative word or relative pronoun. (There is another "that" which is the marker of clausal subordination for that-relative clauses. Some grammars consider that the two markers to be, or could be considered to be, the same. In any case, the "that" marker for relative clauses has a different set of rules in regard to its presence or absence than the one marking declarative content clauses.)
One rather firm rule--well, somewhat a firm "rule"--is that if the declarative content clause is the subject of the main clause, then the "that" marker is obligatory. That is so the reader will get a heads-up to realize that the content clause's subject is NOT the subject of the main clause, even though it is located at the beginning of the sentence.
There are a whole bunch of more similar rules, but they are all rather specific as to the syntactic situation that they are talking about. If the rules were simplified or made too general, then there would be too many so-called exceptions. Native English speakers know these rules implicitly (their ear does all the work), but they would usually be hard pressed to explain them--and if they attempted to explain them, their explanations would often be wrong and/or misleading.