Little update from me, sorry did this trip already a few weeks ago. So long story short, there are various buses, I saw them on the way, if I'm not wrong some bring you from Kunming to the Laotioan border and beyond, others bring you within a day through Laos.
For those of you who want to travel as fast as possible, go to Jinghong and try to catch an early bus to Laos, as some have written here there are buses who leave Jinghong in the morning, which should get you to Huay Xai/Bokeo (int. bus station) within 8 to 10 hours.
For the final leg of your trip into Thailand, there are some direct buses to Chiang Rai, the last one at 5 PM from Bokeo if I'm not mistaken.
On a side note, information provided is very scarce, I heard the trip from Chiang Mai to Luang Namtha with public buses should be very easy and well advertised at the bus station. But especially the other way, coming from Laos is not that straight forward, so you better start your trip well informed, willing to stay over somewhere if needed and with some extra time.

@Alien,

@Alex: maybe I'm wrong, but I think the entire road to Huayxai was built by Chinese companies. In fact the road south from the border to Luang Prabang was originally (? I don't know what was there before) built by the Chinese in the 60s to aid the Pathet Lao & Viet Minh in their war effort (the north of Laos had not been under the control of the central government, or the Americans, for years. It's winding and indeed rather tiring - the road to Huay Xai is much better.

I believe the road from Luang Nam Tha to Huay Xai was a joint effort by Thai and Chinese companies, not merely Chinese ones. Also the 4th Mekong Bridge was a joint Thai/Chinese/Lao effort.

Also, despite the road from Huay Xai to Nam Tha being a joint Thai/Chinese effort and in pretty good condition (I drove between the two towns in 2h45m in my Ford Ranger, Lao registered) whereas years ago it was an 8hour dusty ride by songthaew (pickup with benches at the back), the road strangely does NOT feature any overtaking/passing lanes. No idea why such a twisting and winding road doesn't have such important passing lanes built such that you can safely overtake a slow crawling bus or truck. The roads on the Thai side are a million miles ahead and actually feature regular passing lanes, that is, if they aren't already 4 lanes or more as there are fewer and fewer 2 lane roads left in Thailand.

The Thai/Lao border closes at 2200 (10pm) but for some bizarre reason (possibly due to it's isolation) the China/Lao border closes much earlier, around 5pm Lao time as another poster mentioned.