postponed

  1. The exams have been postponed due to heavy rain, until next month.
  2. The exams have been postponed until next month due to heavy rain.
    Please correct the above.
    Thanks.

I’d use ‘to’ instead of ‘until’ in your sentences.
Alternately, I may say: The exams have been suspended until next month due to heavy rain.

“Postponed until next month” is correct (‘postponed to next month’ is not good).

Sentence 1. is awkward because ‘exams’ and ‘next month’ are separated by the reason for the postponement. It would be better to relate them closer, or to use a separate sentence. I would choose from one of these three options:

Due to heavy rain, the exams have been postponed until next month.
The exams have been postponed due to heavy rain. They will now take place next month.
The exams have been postponed until next month due to heavy rain.

Why? Smaller? Ugly? But I’d prefer ‘to’ in this context.

No, not smaller or ugly. Just not the best preposition to accompany the sentence.

Generally:
‘Postpone something until’ next week, month, year, etc.,
‘postpone something to’ is a reasonable option with a specific date/time or an as yet unspecified date/time which will be specific when it is decided (though you will still find most Westerners choosing ‘until’).

The OLD bears that viewpoint out with its examples:
postpone something to/until something
We’ll have to postpone the meeting until next week.
They have agreed to postpone repayment of the loan to a future unspecified date.

Perhaps you would choose to use ‘to’ because that is what matches ‘deferred’ best.

Not that alone; it matches ‘put off’ as well. The ‘BALD’ may not agree to this.

As a native English speaker, I think ‘until’ is a much better match for ‘put off’.

Oh, another acronym I cannot fathom. What does this one mean?

It’s a word like OLD! The inverted commas suggest that it is figurative. Before I use a new/uncommon acronym, I explain it.

Do you mean ‘bald’ as in ‘hairless head’? If so, I have no idea what you mean by it here, Could you please explain?
If that is not what you mean, then please explain it.