mallyh @15.15 ---we have one like your in-laws. Circular reflective metal round the bulb and has a cord pull. Can have just the light ,heat and light or just the heater. Bought from Argos over 40 years ago. Never had a bit of trouble with it. No need for electric plugs in the bathroom.
When I was working many many years ago, it was in the spec not to have any electrical outlet in outstretched arm to a item like a tap. Kitchen bathroom. Don’t know if it’s changed. Only shaver sockets aloud in bathroom
surely it's not the socket itself that's the danger (modern ones blank off when unplugged) - it's anything that may be plugged into the socket? e.g. I seldom use the toaster in the bath!