There has always been private competition. That's been the case since the foundation of the NHS in 1949. In order to secure the cooperation of the medical profession, Aneurin Bevan, the Health Minister responsible, ensured that the legislation guaranteed the right of doctors to work in private practice, as well as the NHS, if they so chose. That's been the case ever since.
For the first 35 years of the NHS, the private health sector was small, serving niche markets. It was only when structural problems in the NHS started to appear from the mid 1980s that the private insurance industry began to take off, and is now flourishing. Most hospital consultants and other senior doctors work in both sectors.
When I was working at the New Charing Cross (now the Old New Charing Cross a/k/a Imperial College Trust), the ads in the Tube read “It’s a Good Job You’ve Got BUPA”.