We did this trip yesterday...a lovely day and very scenic particularly the first half of the trip.
The public boat is currently leaving Nong Khiau every day at 11am for Luang Prabang. There is also a daily service at 11am to Muang Khua. We were advised to be at the boat office by 10am by which time there was already a queue.
Mr M (M for Mafia is my guess) at the boat ticket office manages proceedings in a most authoritative manner. The boats have maybe four or six comfortable forward facing seats (dining chairs or bus seats in another life?) and the rest of the passengers sit sideways on wooden planks. The trick is to get the good seats but there is no way of eliciting the likelihood from Mr M before parting eith your 110,000kip for the ticket.
We ended up joint forces with four other tourists and chartering a boat for 1,600,000kip. Much discussion ensued in broken English and hand signals with Mr M to be able to view the boat before we paid our money. Just as well we did as there were only 4 seats for 6 passengers. Not a problem to Mr M it seemed but neither we nor our money was going anywhere without comfy seats for all. Eventually two seats were unbolted from another boat and secured on ours and we were happily on our way.
The day was great...beautiful views, villagers along the river and fun avoiding being splashed when going through the rapids (wet weather gear or a towel recommended). The journey took six hours including a drink/toilet stop at a floating restaurant and visit to Pak Ou Caves.