It’s a bit of a toss up over which ex-empire, the British or the Austro-Hungarian, has produced the most number of stamps. Certainly up to WWII the British Empire and its derivatives would be the winner. But in the years since WWII, the countries that made up the former Austro-Hungarian Empire have laid a strong claim to dominance on the strength of 5,000 plus stamp issuing performances from places like Czechoslovakia (and successor states), Yugoslavia (and successor states), Romania, Bulgaria, Albania, and, of course, Hungary and Austria themselves. In terms of interest, the Austrian-Hungarians are winners too. British colonial stamps are largely an amalgam of common design types printed with no regard for the local population in a one size fits all mentality.
The Austro-Hungarian issues retain a nationalistic feel and represent more of what the Austro-Hungarian Empire really was