People from other countries come here to have medical procedures done. In other words I see them coming from countries with socialized health care, I don't see them going to countries with socialized health care.
All kinds of procedures. The medical industry in Mexico is heavily regulated by the Institution Mexicano del Seguro Social and it's quite normal for Americans to cross for surgeries. My family once helped somebody cross for surgery because of his hernia.
Claiming back money from people who come to Britain to take advantage of free healthcare and cheaper medication uses up a load of hospital administration time. Even then it's sometimes impossible to claim back the money from some patients.
The system is there to claim the money back, but it's time consuming and often difficult. More of what I'm saying is that your suggestion that people don't go to countries with socialised healthcare because healthcare in the states is somehow "better" is frankly absurd. Personally I can't think of any reason to go to the states for healthcare when I could stay here and get the same treatment free.
How did they do this? They crossed the border into Mexico. Many doctors in border cities have contacts amongst doctors on the other side of the border. Who paid? They did. Usually, if they have to cross the border, then it means that the insurance wouldn't cover the procedure. The procedure in Mexico is paid by them partly and partly by the government. There is nothing illegal in this on the Mexican side. There is no bribery. That I'm aware of, the US gov't doesn't have a problem with it either.
If the NHS is unable to provide care for you in Britain then they pay for you to have to the treatment in another EU country. The main problem with the NHS which is difficult to resolve is the NHS drugs list. There have been several high profile cases of people asking for certain drugs on the NHS and the NHS refusing because either the drugs haven't been properly tested yet or because it doesn't have noticeable edge on any of the drugs currently on the list.
Edit: I'll just add that obviously the NHS won't pay for aesthetic operations. You'll have to pay for those yourself (I hear Serbia is cheap with an excellent standard of care and expertise).
Healthcare in Mexico is a mixed socialized/private industry. HMOs as we know them in the US don't exist. If a patient is capable of paying for a procedure, then the Mexican government doesn't have to pay a thing. Given the current exchange rate and real prices for medical procedures in Mexico, it's much easier to pay for it across the border than it is in the US.
In the UK there are a certain number of surgeons who can perform a set number of operations each day/week/month. That gives a limited number of people who can get an operation at any given time. The NHS schedules things so that vital operations are performed quickly, with less vital surgeries slotted in later.
If someone is not going to benefit from surgery, it may be bumped back.
The age and weight thing is the same in the USA with regards to transplant lists. People who will get minimal benefit, or are likely to waste the new organs (say, a heavy smoker who won't quit or someone morbidly obese) are not going to be given priority for new organs.