This would be something you would have to pay for privately, under all scenarios barring reconstructive ones.
It's also possible to make a robust argument that where health services are under severe distress, having surgeons, who are often trained in part through public funding, spending their time on cosmetic issues is a really poor way to run things. And that yes, cosmetic surgeries, and maybe even some cases of reconstructive surgery, should be put aside where real health concerns are not a factor.
Additionally, if you have ever done any reading in medical ethics you will know that the ethics of purely cosmetic procedures being done by doctors is an area of significant discussion and disagreement. Many ethicists, and indeed doctors themselves, to not think cosmetic procedures are within the proper remit of the doctors role.
I don't understand why you would think that learning to be content with your body as it is would be something you have a right to avoid, especially on the public purse. No one is obligated to help you get the nose you want.