Also UK, and the problem is basically that there aren't enough ambulances, so they're triaged. It's the same as when you go to A&E and you have to wait hours if they don't think you're an urgent case. Unless you're going to die if they don't get you to the hospital now, they'll put you down the queue (which is fair enough).
As you point out above, the decision to dispatch an ambulance is ultimately the emergency operator's. Typically you tell them what's wrong, and they send out what they think is necessary (which might also be police and fire).
Strange, my family has had need of 3 ambulances in the past 3 years. Never had any issues, other than them taking around 30 mins to arrive, and 2 of the 3 were on Saturday nights.
