In the UK private hospitals and healthcare companies are often little more than parasites piggybacking on existing NHS infrastructure and public investments to cream off some profit. They rarely pay to train their staff (the NHS does most of that), and they cherry-pick the easy profitable things they fancy covering. Many people thinking they're experiencing proper private healthcare because they pay £12 a month to some company they saw advertising on the TV have no clue.
The provision of ambulance services are outsourced to private companies in many areas, following the 2012 Lansley reforms (though they'd started the process back in 2010 in certain regions, re-hiring staff on terrible contracts etc, causing many to quit). This means the trusts no longer have direct control - they're merely 'buying in' a service, and many are frankly a shambles. Coperforma is one particularly notorious operator, owned by the billionaire John Porter. It doesn't really matter that you're speaking to an NHS call handler if the private company providing the ambulances hasn't properly maintained their equipment, lacks ambulances, and hired too few employees in order to save money and boost their profit margin.