It updates instantly, you don't have to wait for the conversation to end. You see it in real-time.
Right now the only two things this doesn't work for are:
1. You don't know your partner started conversation unless they tell you.
2. You are pulled out of the journal to the convo window if a co-op decision is needed.
Other than that, it works just fine for both people to see the entire conversation as it happens.
Right that is why if they are doing a major redo on the UI for it (we don't know) and that will take a good chunk of time.
I'd just take a quick and dirty, give me a hotkey to immediately go to the Dialogs screen. And alert me on screen when the other player starts a conversation and tell me to press the hotkey I set to go read along.
It still isn't the slickest but it gets the job done. I would also suggest when the one player ends the conversation the dialog for the other player reading is alerted, "The End". So they know they can continue on.