NiMH batteries are 1.2-1.25V nominal. There are no 1.5V rechargeables simply because there is no known rechargeable battery chemistry stable, safe and inexpensive enough to bother with, which is why NiCd (obsolete due to toxicity and severe memory effect) and NiMH are the only two that come anywhere near 1.5V.
As for what will or will not run off 1.2V batteries instead of 1.5V, that depends mainly on how voltage-dependent the load is and whether it has a buck/boost converter to get the most out of whatever cells are put in. Some poorly designed toys cut off at 1.2-1.3V, at which point they aren't even using half the total energy in 1.5V cells. Better designed devices will drain their batteries to 0.8-0.9V and should run fine on NiMH.