Are we talking about a corded unit, or a battery one with charger?
If it's corded then it's most unlikely to be suitable for British 240-volt supplies.
However, many (but not all) modern chargers for cordless devices use a type of power supply which will accept a wide range of voltages, so it may be usable "as is." If you look on the unit you should find a label (or molded, stamped equivalent) which gives the details you need.
If it says something like "90 - 250V, 50/60Hz" then it will work here with just a plug adapter. If it's marked "120V 60Hz" then you would need to run via a transformer to drop the voltage.
Does the place you're moving to have a shaver socket? If so, you should be able to plug it into the 115 volts side. You might need to get a little plug thing to put onto the end of your US plug
Shaver outlets will definitely supply enough power for this application. Not all of them have a 120V option though -- some are only 240V. Most of the shaver units which do offer 120V as well have an outlet which will accept a flat-blade American plug without the need for an adapter.