Hello,
People from the US will dial your number as if you were in the US.
UK people will dial 00 1 followed by your 10 digit US number to reach you (this is exactly what you must dial to call back to the US and also what your wife should be using to call you/or you to call her).
To call to UK phones you can dial their numbers using the national dialing format which is 0 + 10 digits (example 07??? ??? ??? for a UK cell). Alternatively I believe you can also drop the starting zero and use +44 7??? ??? ??? which is the international version of a UK cell number.