hopefully

1. Your notion points out a very strong likelihood, WB, but not, I would say
an absolute.

2. CAN seems a more natural choice.

The meeting TO BE held next week ...

Thank you~

Can I say this?:

-I could only hope that the weather would/will turn good.

It's a bit clunky. Try

I only hope the weather will improve.

But do you not think COULD has its function in the sentence?

It might have, in certain circumstances. (quite contrived circumstances imo)

For example,
I told her, in reply to her question, that I could only hope the weather might improve in time for her trip to Brighton.

(That sounds like 19th-c English to me, like something Bronte or Austen might write.)

Thank you~

But do you mean COULD ONLY HOPE itself is old fashioned or only oldfashioned in the sentence you gave?

It is by no means old-fashioned. Using the adverb "only" before the verb "hope" is done to intensify the verb. "We could only hope" means "we very strongly hoped", and could carry the implication that since all other avenues had been exhausted, only hope was left.

If "We could only hope" means "we very strongly hoped", does your answer mean if the act/state of hoping is now, COULD is not possible and CAN should be substituted?

Yes. "Could" is conditional, and refers to the past when used like this.