There are no specific rules relating to the minimum age for passengers. Rather, the pilot is expected to exercise a degree of discretion - based on aircraft type, journey length etc.
The CAA has always proved slightly reluctant to issue advice relating to car seats too. The general advice seems to be that if the seat is approved for Car use, and can be fitted safely in the aircraft, then all is well. This is also the approach taken by the FAA.
One last thought is not to forget ear protection. Kids' ears are much more delicate than adults and need protecting in the noisy environment of an aircraft cabin. Peltor produce a set of ear-defenders called "Kids" (available through the usual pilots shops), which will work for 2-4 year olds. Babies can be a bit more problematic. We took the route of using the Peltor Kids as a clamp, and used terri nappies to pad the ears adequately.
Ours sleep from startup to shutdown too
-2Donkeys