Building a house in America is pretty straightforward process once you have the plans. The plans specify everything. Where the walls are, where the pipes are, where the electricity goes. The foundation type is predicted by the climate. Wait a minute, this is technically automatable. Let me assume a warm climate, like the kind I live in. In this, the house is constructed on a cement slab, onto which are mounted the walls via metal brackets, onto which the roof is mounted. The cold-climate American house has an excavated basement with a cement floor, cinder block walls, and the first floor is built on top of that. Building the wrong type of house for a climate leads to certain....inconveniences.
Given a DXF file of the house plans, the robots should be placed on the plot, and first arrange a coordinate system. This shows them where things are at all times to avoid incidents of robots trampling each other's work, or each other. Robots should also establish where their supplies are, because they are not good at seeing.
Now the work begins as a robot first arranges an area of flattened soil, then builds a wood frame around it. Pipes are arranged in it, according to the placement of bathrooms, laundry rooms, and other heavy water use. The main water line and sewer lines are connected. (In rural areas, the sewer line goes to a leech field and septic tank, but in either case, gotta make the conenctions first. The wood frame is then filled with cement, leaving exposed pipes. This is the foundation.
Now the outer walls are hammered together. The walls are made of wood frames. These wood frames have an upper wood beam and a lower wood beam, each the length of the wall, and support beams at a fixed, periodic distance. The support beams are specified in the local building code, which are laws that specify how buildings must be built so that they don't fall apart. Windows, (the placement of which is in the plan), have shorter support beams up to a four-sided wood box that contains the window. Given standard lengths of wood, the robots can cut and hammer this together. If there are front steps, strong boards should be placed on them to form a temporary ramp, as almost all existing robots move about with treads. (Bipedal motion is insanely difficult for robots.)
The walls are rotated to vertical and metal brackets are placed on strategic places. A masonry nail is driven through the bracket, firmly and permanently attaching the walls to the slab. The corner walls should be attached to each other as well with additional nails, for support. Rectangular slab-boards are nailed into the frame to form solid walls. Inner walls are built similarly, but they are directly nailed with masonry nails without the bracket. Also, no slab-board on the inner walls, as their material comes later.
The roof is made of triangular frames at specified distances, nailed to a rectangular slab-board. The difference between the ceiling and the roof is the attic, a typically unused portion of the house that has the ceiling insulation, and allows the roof to be shaped to repel rain and snow. The roof is, after being firmly constructed, attached to the ceiling with more metal brackets. The house resembles the final house in shape now, and could be lived in if one enjoys living like a medieval peasant. (In that it will keep the rain out and stop the wind, anyway.) The roof will need work to remain water tight, and this is a major priority in regions with frequent rain. In regions with rare rain, we can do this later, so I'll discuss it later.
Now the plumbing needs work. Pipes must be connected to where water-using appliances will need them, and also must have a P-Trap and a roof connection, to avoid sewer gas escaping into the appliances. (Without P-Traps and roof connecting pipes, sewer gas would escape your sink and stink up your house.)
Next, venting. Ducts for air conditioning and/or heating should be installed in the ceiling, and run to where specified in, yes, the plan. Also, any bathroom without a window must have a duct with a fan leading to the outside. Any laundry room will require a duct for the drier to vent steam to the outside. Now is a good time to place these things.
Next, electricity. Electricity flows in wires, which should be run through drilled holes in the support beams. The holes must be placed and sized in ways that do not compromise the structural strength of the beams. The building code specifies where outlets and switches must be, the best place for junction boxes, and so on. Cable should be run in little pipes to protect it, and should meet other cable with a cabling pigtail, in which the ends of the insulation are stripped from the two cables, the exposed ends are twisted together, and a plastic cap is twisted over the bare cable.
At this point, plumbing and electricity inspectors should be sent in for verification. It is best to have this done while the two are both exposed, as any mistakes require us to pull out the defecting parts and replace them. This is easier to do when one does not have to rip through the wall to do so. The robots should line insulation at this point around the edges of the house, and install the windows, two tasks that will not interfere with the human inspectors.
When the plumbing and electricity are approved. it's time to make the inner walls more...solid. Sheets of drywall are nailed to both sides of the inner wall, then a spackle covers the nails, and is allowed to dry. These walls are ugly, so they should be painted. Robots should only paint the priming layer, which is typically white, because I'm not about to trust a robot's sense of color. If a color is already specified, they can paint that after the primer dries.
Now, flooring. The floor currently is just a cement slab, irregularly grey in color. Most people find it drab. We can, depending on the room, install carpet, wood, or tile, to make it more pleasant to look at and walk upon. For carpet, a "tack strip" of sharp hooks is nailed to the edges of the room, carpet is unrolled from sheets and pulled tight. Cut excess carpet. Robots can do this because the specification shows the exact size and shape of the room.
For wood, robots must pick an arbitrary starting point, lay out a plastic "underlayer," and then nail wood to one edge until it is full. Now further boards attach to the boards on this edge, filling out until the room is covered in wood. Flooring wood typically comes stained, treated, and so on, which is a good thing because those steps would be a major pain in the ass.
For tile, robots pick an arbitrary starting point, and cut any tile that would not fit in its assigned space. They then obtain a mortar mix, and slather the back of a tile with mortar. The tile is placed in its assigned place. Regular small gaps are placed in the tile, as we humans like it that way. When all tiles are placed, a gritty grout is rubbed over the tiles, most of which fit between the regular gaps. When this is all half-dry, the surface is wiped, removing excess grout.
Kitchens and bathroom sinks require some cabinetry. Robots know the required shape from the plans, and cabinets are made of wooden boxes, held together by nails. Bathroom cabinets and the kitchen sink cabinet need holes drilled for the sinks, which the robots should have on reference. Countertop in both room has two options. A solid slab, precut and lifted into place, or tiles, which are made exactly like floor tiles. Countertop should be in place before cutting. The sink is attached to the cabinet via a silicon goo.
Some standard appliances like toilets, bathtubs, and maybe refrigerators should also be installed now. They are lifted into place, attached with bolts, and connected to the water supply with connectors laid down on the plumbing stage. (I'll trust the humans to plug the refrigerator in. Robots could do it, but it's less urgent.)
Now, roofing. This should be done before attaching the roof in a rain-risky climate, and is probably easier for the robots before attaching it to the ceiling anyway. First, a water-proof plasticy layer is stapled to the wood sheet. Then, tiles made of asphalt or ceramic are stapled to the plastic, starting at the bottom of the roof and working upwards. The tiles must overlap each other, and they are the main defense against rain. At points where a pipe sticks up, such as the ones for the plumbing or roof venting or air conditioner, metal flashing should be placed around the pipe, and roofing tile layed over that. (Cut the tiles as needed.)
Houses need a facade now, because the exposed insulation looks vaugely ridiculous to the average home buyer. Many options are available, but robots should select one (perhaps specified in pre-construction specifications?) and install it on the exterior of the house. There is aluminum siding, wood siding, vinyl siding, and fake brick siding. All of them install in strips.
All of the construction has probably damaged any plants in the area beyond repair. Wood-hammering robots should construct a fence, and other robots should lay down strips of grass sod. When watered, the grass lays down roots and becomes a lawn. The house looks complete at this point.
At this point, the robot's work is done. They should return to their packed state and be ready to be shipped off for the next house to construct. Two more humans must see to this house before we sell it. One, an interior decorator will want to place things like attractive curtains, and choose colors for yet unpainted surface. (The decorator will not do the painting, but will describe the proper color, which can be done by robot or human painters.) Also, local government will want to send down a tax assessor, to determine what property taxes the owner of this building needs to pay. The temporary ramp should now be removed.
We then put this house up for sale, and hopefully get a buyer fairly quickly.
Now of course, this post is so vague that it would drive any real engineer insane, but I think this is a good starting point for this sort of idea.