In the majority of UK-based traditional boiler setups, what does the hot water thermostat look like?
A boiler cylinder thermostat is usually a small rectangular box, often with a temperature adjustment knob (which may be recessed for screwdriver adjustment), which is fixed to the side of the cylinder with a strap or cord going round the cylinder horizontally. Sometimes this is an elastic wire like net-curtain fixing wire. The insulation between the thermostat and the copper cylinder should be removed so the thermostat is in good contact with the cylinder metal. It will have an electric cable coming out of it to a junction box connecting to the boiler control circuits. 

In some cylinders the thermostat will have a probe on the back and this will slide into a pocket in the cylinder to make good thermal contact. 

If the thermal contact between the cylinder and the thermostat is poor, the boiler will not recognise the water getting hot enough and your hot water will be too hot.

The other thing(s) actually on the cylinder will be immersion heaters, and they will usually be at the very top or the very bottom of the cylinder, be fixed into bosses on the cylinder, and have a stiffer, thicker, electric cable to a switch on the wall. 

The immersion heaters will have thermostats under the covers to control the water temp when using the immersion heater, but this won't affect using the boiler. 

Older boiler installations often don't have a cylinder thermostat and rely on the boiler's internal thermostat to control heating and hot water temperature, which is wasteful. 

Larger boxes connected to the pipes are likely to be zone valves to change-over between heating and hot water. They'll thunk when they operate, and will often have a manual override lever on the side.