Learn English – While I was there

Using "need" like that is a little old-fashioned, but still widely used. It comes off as very proper/fancy.

Since you say in the title that you want to tell the person that there was no need to (ever) worry, you need the past tense.

The most obvious and native-sounding way of saying this to me would be:

"All the work was completed on time, so you had nothing to worry about."

It is a bit idiomatic as "nothing to worry about" is a very common phrase. It sounds very natural.

If you want to use "need" specifically, I would say:

"All the work was completed on time, so there was no need to worry"

Saying "he need not worry" is perfectly grammatical and easy to understand. It just sounds very formal. Most people would just say "he doesn't need to worry." You can read more about that here.

There is maybe some syntactically valid way of phrasing it like "so you need not to have worried" but it sounds very awkward to me, and it takes some effort to understand someone saying that.

"I'll be there in an hour" denotes arrival in one hour from the time the words are said. "An hour" is also generally used to be a bit vague, whereas "I'll be there in one hour" is generally used to be more specific. Either way, the clock starts from the time it's stated.

"I'll be there one hour later" implies that it will be later than something, i.e. one hour later than already agreed. If you'd agreed to meet at 15:00, this means you'll be arriving at 16:00 instead. You could say "I'll be an hour late" to convey a similar meaning.