Such a timely manner answer!
is so unnatural as to be wrong. i have never heard a fluent speaker say anything of the sort.
The form I believe came first was:
He answered her in a timely manner.
Here "timely manner" functions as an adverbial phrase modifying "answered".
Your other suggestion of
This answer came out in a timely manner.
again has "timely manner" qualifying or describing the action of 'came out".
One could say:
The answer was timely.
Such a timely answer.
but in either case I think "prompt" or "quick" or "rapid" would be better than "timely". Strictly speaking, "timely" means "at a particularly appropriate time" rather than "quickly" or "early". For example:
The delivery was timely, we needed the item only five minutes later.
Of course, early or quick response often is appropriate, and so "timely" can often be used for things that are prompt.
Whenever "timely manner" is appropriate, "timely fashion" or "timely way" can be used instead, with no change in meaning.