Learn Turkish: Çorba neredeyse hazır; beş dakika daha bekle. - The soup is almost ready

Yes, but it changes the meaning. Using hazır without the suffix means you know for a fact the soup is almost ready, likely because you are the one cooking it. If you added the epistemic suffix to make it hazırdır, it would mean 'The soup is probably almost ready' or 'I assume it is almost ready' (for example, if you are sitting at the table and making a guess based on the time).

How does 'neredeyse' come to mean 'almost'?

It is a combination of nere (where), the locative suffix -de (at), and the conditional suffix -(y)se (if it is). Literally, it translates to something like 'wherever it is at'. Over time, this phrase merged into a single set word that Turkish speakers use to mean 'almost' or 'nearly'.

Why is the word order 'beş dakika daha' instead of 'daha beş dakika'?

When daha means 'an additional amount' (like 'five more minutes' or 'one more tea'), it comes after the quantity and the noun: beş dakika daha. If you place it before the time phrase, like daha beş dakika, it changes the meaning to 'still' or 'yet', as in 'There are still five minutes left'.

Why doesn't the verb 'bekle' have a time or personal suffix?

Bekle is the bare verb stem, which is used as the informal, singular imperative (command) form. It simply means 'wait!'. Since you are likely talking to a friend or family member if you are cooking for them, this direct command is perfectly natural and not considered rude.