Getting Back On Topic
Bringing a Euro/UK washing machine to the United States will depend, as another poster stated on how much and what type of power is required, and what is being powered as well.
You have already found out that most European wiring is 220v/240v single phase (that is one wire brings in 220v), versus the United States system of two wires each at 120v but out of phase with each other. There is also the matter European/UK power is at 50hz and the United States at 60hz.
Older washing machines and other appliances with simple electronics and or mechanical systems might not be bothered with the switch, provided one hooks them up carefully. However modern washing machines with computers/motherboards probably will not work, or work very well because the computers are set for 50hz, running them at 60hz can cause all sorts of problems ranging from faster cycles to increased spin speeds.
To run an appliance that needs single phase 220v power exclusivly, the simple solution would be to use a step-up converter rated at least 50% more than the maxium power draw of the appliance. That is to say if the washer draws 2500w, the coverter should be rated at least 5000w. Converters however will not change the frequency (hz) of the power, thus if the appliance truly needs 50hz versus 60hz, you will need a box to change frequency in addition to the converter, these are dear and quite large.
Converters for use in the United States generally stop at 3000watts, which is slighlty over the max power that can be safely drawn (by code) from a 20amp electrical circut in the United States. While you can find converters at 4000w, 5000w and up to 10,000w, it is not generally safe draw that much power from a 20amp circut for a long period of time. The last two converters are almost always sold as units which must be hardwired into the home's electrical system.
Being as all the above may, there are/were several persons who purchased AEG washing machines either from a source in Canada (Euro-line appliances), which run on European 220v/50hz power. IIRC Euro-line gave instructions upon shipment on how to make the units work with American power. IIRC, one person even ran his off a converter. Occasionally one sees other European washers and or dryers for sale in the United States, some apparently used here, which means the owner's found a way to make them run.
In regards to power draw; if you do not intend to use the washing machine's heaters often or for long periods of time, you *might* get away with using the washer with a converter. That is to say if one sets the machine to "tap cold" and uses whatever water comes into the washer, and or starts with water say at 120F and heats it to 140F, pulling 2400 watts for a short period of time may not cause damage, however I'd speak to an electrican and have him see what sort of condition your wiring/outlets are in before considering such a move. Appliances such as washing machines pull their max power when heating water, usually their motors are 90watts or less, then there is some power to run other systems, but again they are much less than the 2500w or so used to power the heating elements. Even the max draw of the heating elements is only a short part of the cycle (depending upon incoming water temp versus desired water temp).
You would need a dedicated 20amp circut, with proper wiring from the panel to outlet, not merely a 20amp breaker/fuse installed in the panel and outlet changed.
Personally unless the washing machine in question is so extraordinary to warrant such time and effort, I wouldn't bother. You can find many good vintage Miele washing machines in the United States that run on American 220v, that will do boil and very hot washes, the main reason for all that power.