Can I ask your nationality, scourge, so I can ascertain what you mean by "things we know" about universal coverage?
Wait times vary from procedure to procedure. Urgent procedures are performed as soon as space for one can be found on the theater rosters (the two relatives of mine who were unfortunately stricken with cancerous tumuours had surgery within a week of their discovery when such was possible without the prior need for chemo/radio). Wait times for non-essential procedures (those not required to preserve life or a basic standard) like cosmetic surgery tend to be a great deal longer.
Referral to specialist departments tends to take around a week to two weeks at the most if one is considered at risk of having a serious condition, and two weeks to a month if one is suspected to have a more minor condition (for example, things like acid reflux or mild allergies.) The NHS is committed to reducing waiting times for all surgeries, and so far they're on target in this aim. Waiting times for all surgical procedures are declining, as are the waiting times for referral appointments with specialists, with the exception of psychaitry referrals.
Bureaucracy? The doctor diagnoses us, we're referred onto the appropraite specialist if necessary, or treated in his office with presriptions and minor procedures. NHS patients face very little bureaucracy while receiving care.
Co-pays? Healthy working adults pay for their own medications, unless usch medcations are given as part of in-patient hospital treatments. The charge for prescriptions is fixed at £5 in England, £4 in Scotland, regardless of the item prescribed. Healthy working adults also pay for their own spectacles and dentistry. Children, the unemployed, pregnant women, new mothers, the retired, the disabled and full time students pay nothing.
Denial? The only way you'll be denied any treatment is you're not considered strong enough to withstand it (i.e. some surgeons refuse to perform non-essential surgery on the extremely morbidly obese due to the risks posed to them by the anesthetic) or if a specialist over-rules your general doctor and decides that medically, you've been misdiagnosed or would benefit more from a different treatment.