Lean-tos are hard-sided structures, and are not really meant to be set up quickly. As a hiker and scouter, I and my group carry tarps all the time, and expect rain to greet us wherever we decide to set up camp. As such, tarps need to be one of the most accessible things you have.
If I were caught in a downpour, my immediate reaction to get/stay dry as quickly as possible would be to generally follow these steps, which does require a leader to step up assertively, and to have knowledgeable people in the group. When either cannot be accommodated, your chances of a quick setup will look bleak. And wet.
First thing occurs before you even set off: You must pack appropriately, and everyone must be aware of where everything is. This isn't just for good measure to stay dry, this applies in cases of emergency. So, staying dry, staying warm, and having fast access to first aid kits, maps, and means for calling for help need to be very accessible, and everyone has an interest in knowing where these resources are. Whether you are car camping, hiking, or whatever, everyone ought to know where the emergency equipment is. Everyone should have personal rain gear.
Next is when it begins to downpour. The leader must take charge: Assign someone the various tasks, such as retrieval of rope, tarp, stakes, etc. Not related specifically to your situation, but anyone who doesn't have a job needs to don rain gear. Staying dry is a safety issue, so, everyone needs to be as dry as possible.
First tie a rope across two trees, as high as possible.
Next, toss a tarp over it, and guy the tarp with lines, secured to trees, rocks, or pegs.
While tying out the tarp, everyone not doing anything needs to remain under the tarp.
While at a Klondike event this past March with the boy scouts, the camp winners did this in a breathtaking 38 seconds. Their method cannot apply to everyone, and I suspect that they got wind of the specifics of the event and then practiced it. But here's what they did:
Their patrol leader assigned 4 scouts to take on the rope across two trees; one scout climbs the back of another to gain height; each pair ties clove hitches to the tree using 1/2" rope.
As the scouts get into piggy back, and another lays the rope on the ground between the pairs, another scout is assigned to retrieve the tarp. Rather than throw the tarp over a tied and taught rope, they performed the novel task of throwing the tarp over the rope as it lay on the ground as the pairs of boys assumed their piggy-back position.
Once the tarp was laid over the rope, and the pairs of boys were piggy-backed, the other scouts handed the pairs the rope, which was first secured on one side, then the other; meanwhile, others were tying the tarp corners with rope. By the time the tarp was raised and the clove hitches were being secure, the guy lines were being secure.
38 seconds was the record for the day. Group of 8 worked as one well-oiled machine, very impressive. I definitely took notes!
This was no easy feat. Tying the clove hitches on the right side of both trees, getting that one scout to fetch the tarp and unfold it across the outstretched rope, getting the other scouts to tie 2-half hitches on all the corner grommets, getting a piggy-back, raising the center rope, tying off the guys... that took practice.
I wouldn't expect every camper in every group to rehearse something like this, but each group needs to have the know-how and the equipment to efficiently set this up. The key is the leadership to properly communicate what is needed, and the others to know how to perform their task.