The radiators are installed under the windows because, as someone else already mentioned, the windows are thermal bridges in the room.
Since the "cold" air that penetrates through the windows falls downwards, it is guided through the grilles above the radiator, where it is heated and then exits below as warm air that rises again in the room.
No offense intended, just so this statement is not adopted by others: No.
1. Radiators have always been and still are installed somewhere on the exterior wall, exceptions do occur. This is because the building’s transmission heat losses naturally occur at the building envelope, and the temperature would be lower at radiators mounted on interior walls. This would lead to drafts, condensation due to cold walls, and other problems. In new buildings with exterior walls that have high thermal insulation, this is no longer as important but still helpful. Underfloor heating avoids this by providing even heat distribution throughout the room, working better the higher the thermal protection of the building envelope is.
2. Windows are not thermal bridges (nor cold bridges). Thermal bridges are something else. For example, there are geometry-related thermal bridges, like building exterior corners, or construction-related thermal bridges, such as connection points from interior walls or roof to wall, etc. Windows are simply components which normally (at the same energy standard) have a higher U-value than the opaque exterior wall. This results in higher heat loss and a lower surface temperature on the interior side of the component. In older windows (older than 1994), the U-value was so high that the indoor air cooled significantly at that spot and, due to the higher density of the cold air, it basically "fell" downward there. It is not the case that a lot of cold air penetrates there, unless you have a (stupid) window frame ventilator or very old windows and it is windy.
3. The air in radiators does not "fall" from top to bottom and is heated in between. Convection heaters / radiators use the chimney effect for heat emission, which occurs when the air that is rather cold near the floor is heated in the spaces within the radiator, expands, and rises due to its lower density (hot-air balloon?). The resulting underpressure at the lower inlet of the radiator causes more air to be "sucked in" from below -> natural convection on hot surfaces with chimney effect.
If radiators are placed under windows, the effect of the air cooling and sinking at the windows is offset by the warm air rising from the radiators and ... tadaa no drafts, no cold surfaces, but dry windows due to the warm air passing along and an even temperature distribution in the room.
With newer windows, the heat losses are significantly lower (less than one third of the "old" windows). As a result, this effect hardly occurs anymore and underfloor heating is possible without restrictions and radiators can be placed elsewhere.
By the way, the typical temperatures of underfloor heating in new buildings of 30-35°C cannot be achieved with classic radiators. Convection radiators require a temperature difference to the room air of about 15-20°C to utilize the chimney effect; below that, they mainly function through heat emission by radiation on the surface. For the load case, they would have to be massively oversized if you cannot use the convection effect with the low temperatures.
Therefore, well-designed radiators require about 45°C in the supply line and 35°C in the return line. Alternatively, the radiators can of course be actively ventilated, similar to how air/air heat pumps work – with the same comfort downsides. But then heating circuit temperatures below 40°C can also be realized without underfloor heating.
