There can be occasional waits for NHS treatment, but for most people the experience is excellent most of the time. For example, when I went to casualty in early January I was seen withing half an hour, X-rayed 20 minutes later, and out of the hospital, knowing I was in the system for follow-ups within an hour and a quarter. My follow up visits have been well organised and efficient.
On the other hand, I have known of incidents where people have not had necessary treatment, incorrect treatment, or have been forgotten by the system. In one case it was because of a pig- headed consultant who could not admit that he was wrong. Another incident was caused by a doctor who changed a friend's medication (probably to a cheaper alternative) and destabilised his condition. A third incident was a GP surgery trying to shuffle off costs onto the local hospital, resulting in a paperwork mess.
Where they do have problems is when treatment is seen as non- urgent. This can include plastic surgery for those damaged in accidents, or treatment that could improve quality of life for a patient, but would be expensive in the short term. Such treatment can be delayed for years, simply by putting up barriers to such people entering the system.
Much of this is caused by the rise in the cost of treatment as technology and medicine develops. But one result of this has been that the government has spent billions on reorganisations and business "experts" instead of on healthcare. In particular, the semi- privatisation of many hospitals has led to a ballooning of administrative costs which are contracted to private (often American) companies for up to 30 years ahead.
No system is perfect. The British NHS works almost all the time, even for very poor people, which is a big plus. It is hampered by the enduring British desire to get something for nothing. Although our health service is much cheaper than most alternative models, and voters consistently claim in opinion polls that they would like to spend more on it, when it comes to the ballot box they will almost always take the tax- cut bribe.