What effect does the body have on the character and quality of images?

After you eliminate "... things like APS-C vs. full-frame, in-body image stabilisation, ISO range or resolution..." there's not much left. Sensor size, pixel size, sensor efficiency (how much of the light that hits a sensor actually makes it down a pixel well and gets counted), etc. are the primary determining factors in how images of the same scene recorded through the same lens by various sensors differ. Sensor design decisions for things such as on-die noise reduction or post analog-to-digital conversion noise reduction will also affect things to a degree. On-die NR tends to "eat stars" by eliminating weaker stars along with noise in cameras that do more noise reduction on the analog signal before it is converted to digital. They also give cleaner shadows when the shadows are pushed, for the same reason.

When people say things like Canon cameras tend to have warmer skin tones while Nikon's skin tones are cooler they're really referring to decisions made with regard to the way the camera's firmware converts the raw data coming off the sensor into color information.