The usual or typical wording here would most likely be:
The future has not yet begun.
Just like: The movie has not yet begun. The play has not yet begun. By creating other sentences with the pattern, it's obvious.
yet limits the verb to the time of speaking.
If something has "not yet begun", it can't be paired with never. Only with "never yet" before the verb.
The future never begins when I want it tooto.
The future has not yet begun or begun yet.
The future has never been easy to face.
The play has never begun on time. The play has not begun yet or yet begun.
Can you say: The play has never begun yet? The story has never begun yet? No, those are not grammatical because they don't make sense.
You have to choose: The play has never begun on time. The play has not begun yet. The play has never yet begun on time. [adverb: until now]
If something is never, it can't simultaneously be yet with regard to the verb. Except in utterances like, where it is adjectival or pre-positioned adverbially:
A never-yet-seen scandal.
Or pre-positioned to the verb:
The future has never yet begun when we wish it had.