I've spent a major portion of my career as an engineer working in and on hydroelectric generating plants of a wide variety of types/sizes. I've also been around a few waterwheels over the years.
A wooden waterwheel, while picturesque, is one of the most inefficient ways to use water a power source, even if that "fuel is free". You do not say whether your site lends itself to an overshot or undershot wheel. If undershot, then a wooden wheel would be a pretty good choice. If overshot, a higher efficiency wheel with curved buckets such as the "Fitz" wheel would be the ticket.
Most waterwheel driven mills used some form of "speed increaser" gearing between the wheel shaft and the mill's lineshafting. Waterwheels turn quite slowly. Usually, a cast iron bull gear was mounted on the wheel shaft or on the wheel itself. This bull gear turned a much smaller pinion on the first shaft into the mill. Something like about 1:15 speed increase was typical. The bull gears were usually not machined and ran "as cast" teeth against the pinion. Plenty of gear lash and plenty of grease and this arrangement worked for generations of people without needing any parts replaced.
Most of the waterwheel driven mills did not bother with a governor. Rather, the miller or sawyer (if a sawmill) had a handwheel or lever to open or close the gate on the flume or sluice supplying water to the wheel. As load came on the machinery in the mill, the gate was opened as needed and then throttled down or closed when the job was done with. A lot of sawyers and millers went as much by the sound of the saw or millstones and "feel" of how things were running.
A waterwheel has a lot of stored inertia, so once turning, however slowly it might be, the wheel tended to even out the rpm a bit when load changes were small. Something like sawing a log in a sawmill would pull down the rpm and the sawyer would open up on the flume gate as the saw lugged down.
The bearings for the old mill wheels were usually babbitted pillow block bearings (plummer blocks in the UK). Some of the old mills ran hardwood bearings for the wheel shaft and also for some of the low speed shafting. I've done a bit of engineering work with a shop that builds re-creations of old mills and repairs existing ones, and have seen quite a few different designs. The other piece of "elegant engineering" was the use of cast iron gearing with inserted hardwood teeth. Typically, a low speed gear would be made up using this combination, and the hardwood teeth would run on the teeth of an "as cast" pinion gear. It made a long wearing and quiet set of gears. Often, a set of moulder knives would be made up to mill out a length of hardwood for "gear tooth stock". Teeth were then sawn off and inserted in the cast iron gear body.
On a few of the Fitz wheel jobs I was involved in, we used roller chain and sprockets to connect the wheel shaft to a jackshaft and increase the speed. From the jackshaft to the induction generators, we used timing belts. These were Fitz wheels made of steel plate, and were erected in some town parks where there was existing water flow and head to run them. The wheels ran pretty much unattended, and the induction generators "floated in the grid" and pumped in a little power to offset the meter reading for lighting of the parks and ballfields.
If you build a wheel, you will need to build some sort of sluice (open flow channel) to direct water onto the wheel. You will also need to build a sluice gate to control or shutoff the flow of water to the wheel. A simple sliding gate made of wood is adequate, but allow for the swelling of the wood gate in the guides or grooves. Often, the sluice gate was worked by a simple lever arrangement, made of wood, and the lever could be pinned in a number of positions to set gate openings for given loads and heads. The other variable with any hydro plant or water wheel is head. If the plant is "run of the river", such as a dam on a river of consistent level (barring flood conditions, spring freshet, or drought), the head behind the dam is usually fairly consistent and some spillage over the crest of the dam is typical. On a "run of river" type situation, a wheel could run continuously. If the mill is built on a mill pond fed by smaller streams, then the "peaking and ponding" mode of operation is used. Typically, the wheel is not run at night, and this allows the streams to fill the pond up so there is ample water and head to run the wheel the next day. As the day progresses, the pond level is pulled down by the flow thru the wheel, and more gate opening is needed (with more flow at the lower head) to make the power needed to run the mill or do whatever work is required.
Water power is quite interesting, and there are extreme times when a river goes to flood stage, or ice jams break loose, or a log jam of floating debris clogs the forebay of a plant. Then, there are such niceties as various stuff like water chestnuts and grasses that come downstream to block or obstruct flow into a mill or hydro plant. Been there, dealt with all of the above. A trash boom goes a long way to keeping floating debris (limbs from trees, whole trees uprooted by flooding, basketballs, old tires, plastic jugs, chunks of rigid foam that broke loose from someone's floating dock upstream...) out of the headrace.
