Most people I know still reserve the word régime for government. I have not found the crossover in Cda that you seem to have in U.K.

For my own personal edification I looked it up in the Websters New Collegiate Dictionary. In the U.S. the words do share the same the <systematic plan, as of diet etc.> definition.

Perhaps regime is gaining acceptance as a course of therapy/diet because most of these diets are faddish. Fads tend to adopt a language that makes them seem more 'real' or 'official'. Régime is French for diet and if you are going to market a diet (regimen) it is an easier sell if you use a word that sounds more exotic in its name. Eau de toilette sounds so much better than the real description of the product (watered down perfume oil). Citrus régime sounds better than grapefruit diet. Marketing is good at blurring the truth (or making it seem a little less harsh.) A régime sounds so much easier to take than a diet - et voila - a new definition for a word is accepted.