I think it depends where in the UK you are. In Scotland, if you're under 26 I think you should have access to free dental care. I'm not sure about the rest.
However, as taken straight from the NHS website:
"NHS orthodontic treatment is free for people under the age of 18 with a clear health need for treatment. But because of high demand, there can be a long waiting list.
Orthodontic treatment is not usually available on the NHS for adults. Most adults have private treatment.
If you do not qualify for free NHS treatment, or you do not want to wait for treatment to start, you may choose to have private treatment.
Private orthodontic treatment is widely available, but expensive. The fee can range from £2,000 to £6,000, depending on the complexity of the treatment and the type of appliances used, but fees can be higher.
After an initial assessment, a private orthodontist will talk to you about a possible treatment plan, how much it will cost and any alternative options you have."
If the dentist considers your request to be of a mostly aesthetic nature, and if no procedure is done it won't have a negative impact on your dental health, you would have to go private to get the issue addressed.
As someone who went this route, don't be put off enquiring at a private practice - it is expensive but I would avoid postal services like 'Smile Direct Club' because you won't have a dentist there with you in-person ensuring that things are progressing correctly.