I will be in the USA for a few shows. I hope to use my Hall of Fame & my Flashback X4. Will they work properly over there with the power supply I use here in the U.K.?
Yes, it will work fine (aside from the differing plug-type, of course!.
Most of today's power supplies will automatically adapt to different electric currents all over the world. The Flashback X4 includes a power supply which handles mains electricity from 100 to 240V and frequency of 50/60Hz.
The only problem is the difference in electric outlets and plug types. The power supply included with the X4 series, however has different plug adapters for USA, Japan and Europe and so you should of course change to US plug variant.