Solution 1:

The usage of mad to mean "insane" is considered a Briticism in America—go figure. Yet mad is actually a very diverse word. As an adjective, mad can mean insane, furious, rabid, wild, frantic, even hilarious. The use of mad to mean "beside oneself with anger" dates back to the 14th century. Rev. John Witherspoon, a Scottish-born signatory of the Declaration of Independence, dismissed that usage as an "Americanism" in 1781. I guess we Americans have ourselves to blame.

Solution 2:

I would say the British can use Mad as a synonym for angry - at least locally:

http://www.oed.com/view/Entry/112000?rskey=iPIgLa&result=5&isAdvanced=false#eid

b. Angry, irate, cross. Also, in weakened sense: annoyed, exasperated (with †against, at, with, etc.). Now colloq. (chiefly N. Amer.) and Brit. regional.

Solution 3:

As a native Brit, I frequently use Mad to mean both Angry and Insane... particularly when I am at work(!).

I don't think there is much of a trans-Atlantic distinction in meaning.