Outlet Problems
The first problem you will face is that your U.K. Wii has a power socket that does not fit with U.S. outlets. This, however, is quite easy to solve. Most electronic stores--try the shops at airports aimed at travelers--carry a variety of inexpensive outlet converters.
Voltage Problems
Another issue is that U.K. and U.S. outlets work at different voltages or power levels. While it is possible that your Wii might work when plugged into an American outlet, this is not recommended as it has the potential to damage your system. To solve this problem, you need a 110/120 to 230/240 "step-up" transformer which converts the American 120 voltage outlets to be able to handle 240 voltage devices. These converters are fairly inexpensive and available at most electronics stores.
Before buying an outlet converter, take a look at the transformer you are planning to purchase, as most have a plug converter built in.
Game Region Lockout
In some parts of the world you also could face game region lockout. Games for consoles are sold to be used in specific parts of the world. Your U.K. Wii is built to play European region games and cannot play American region games (or Japanese region games). So, you will need to play the appropriate games for the consoles.
Getting Around Region Lockout
There are several ways to get around the region lockout on Wii games. Most require physically altering your system and will void your warranty. If you really must play U.S. games on your U.K. Wii, however, there is a product called Wii FreeLoader which gets around region lockout. Keep in mind, however, that it is not an official Nintendo product and you use it at your own risk.
Wii Internet Connection
You do not need to worry about connecting a U.K. Wii to the internet in the United States. There should be no problems.