A term used in a time of despair, anger or frustration. often associated with annoying twats.

used to describe an instance of frusteration, annoyance, or grief.

the same shit that for fuck sake but with different timing

For the sake of all the fucks.

Variant: For fuck's sake. (For the sake of a specific fuck).

for the sake of something
stated when confused, surprised, or when annoyed

A swear that dates all the way back to 1800.
It is rarely used nowadays sadly.

How the Scottish and Irish spell and pronounce "fuck's sake", or "for fuck's sake".

We don't say that, we just say "fuck sake". Why that would be is anyone's guess, probably just another of our numerous attempts to butcher the English language, which is a fine practice. The likelihood of an entity called "sake" that we are opposed to grows smaller over the years, as far as I can see.