Not exactly. The Crown is the main thing holding it all together but, because of the way the UK is run, there is legislation covering us. I think the link provided earlier goes into it.

But if we did decide that we no longer wanted to be subjects the monarchy it would represent a massive constitutional crisis for us because our entire system is set up for that. I'm not even sure we could without royal permission. Though I'm sure in today's political climate we would need only have a referendum and then ask (if it was popular) and it would be granted. Then would begin the process of uncoupling ourselves from the UK internationally, we would have to join the UN, WTO and other international bodies we are not currently a part of as it's all handled by the UK.

It's not as simple as just saying "we aren't a BCD anymore".

The day a firstborn daughter ascended the throne of the UK and her brother ascended yours, you'd lose the protection of the UK and their administration, you'd cease to share citizenship with the UK, etc?

Such a thing wouldn't be possible. We dont get to choose who in the royal family is our monarch any more than the English do. We can't choose to have a different one to England or the rest of the Commonwealth.

If we wanted to do that, we would have to become independent entirely, and then choose a path of a monarchy and choose someone from our own population. Though that's moot because I imagine if we went through all that trouble we would just pursue federalist or republican democracy.