It's very rare for hair dryers, curlers, etc. to be dual-voltage, so any regular British appliance of this type will almost certainly be wired for 240 volts only, and thus not suitable for running on a normal 120V American outlet with just a travel plug adapter.
However, most American houses do have 240V available in them as well as 120V. It's commonly used to run the range, clothes dryer, air-conditioning, heating, etc. (not regular domestic refrigerators though -- they are 120V).
If you can find a 240-volt 15-amp outlet, you can run from that by changing the plug on the appliance or using a suitable adapter lead. You might be able to find one of these 240V outlets feeding a window air-conditioner, a larger baseboard heater, or something similar. You'll need to check each such possibility individually though, because the smaller units can be just regular 120V.
The 240V 15A outlet is similar in overall size to a normal 120V receptacle, but has the two flat blades in tandem instead of parallel, like this:
If you can find an outlet which looks like that, it will be 240 volts.
