The British monarch handed it's power over to parliament.

This isn't really what happened, there was a roughly 700 year period of time where power was contested between the monarchy, the nobility, and later representatives of increasingly larger shares of the non-noble populations.

There are also a whole bunch of other European countries that have gone down similar paths and ended up in a similar place, the largest of which being Belgium, the Netherlands, Denmark, Sweden, Spain, and Norway.

Japan also has a constitutional monarchy, but in Japan the transition was rather abrupt in the aftermath of WW2. While the Japanese Emperor had long been considered semi-divine, they had virtually no real power for most of the past several centuries, with the real power being held by the military, bureaucracy, or warlords known as Shoguns.