Can't make up my mind
It was (and is) indeed a constitutional monarchy, but it was (and is) also a democracy. That's not to say it's a democratic republic, like France or the US, but the fact of monarchy in the United Kingdom is that it now largely serves the role of figurehead and rallying point -- the United Kingdom is fully democratic, and lots of people (myself included) think it's the oldest democracy in the world. Obviously, this point is debated between Britain, France, Switzerland and the US, who all claim to be the oldest, owing to their different definitions of democracy -- but there is at least a certain for the claim that the UK was a democracy during the time of empire.
As for that term -- empire -- you're right in thinking that the British Monarchy only started calling itself an Imperial bloodline after India in 1857, styling themselves Emperor or Empress of India, etc. etc., but the political institution of empire has a much longer history in Britain.
Again depending on definition, you could reliably argue for the existence of the British Empire since the Acts of Union in 1707 brought de facto sovereignty of the English Parliament over the Scottish Parliament.
Overseas empire far precedes this date, however, as the English (and then British) were establishing colonies in the Americas since the Chesapeake Bay Colony, Jamestown, Williamsburg, etc. (You know Pocahontas and all, eh?)
So, to wind up an increasingly long and pedantic post, Britain was an empire long before the monarch started calling him/herself Emperor or Empress, and India was only the most lucrative of British Imperial conquests, certainly not the first!