I find the comments here harsh.
My sister has massive boobs, she is getting them reduced on the NHS due to the weight, pain and the mental distress.
If she can get them reduced, then I can't see why the OP if she has an actual clinical problem, can't get them augmented. It is causing her mental distress.
Attitudes like many female posters here is why the NHS is so shit at dealing with womens physical, sexual and mental health issues. Smile with the bad service!
I can tell you my ex went to the doctor because he couldn't get his penis to rise and by god, they pulled out all the stops to help him get an erection. Oh that's because it's crucial for men's self esteem. But women can suffer crippling endometriosis and suffer without diagnosis or have no sex drive or have no breasts at all and they get told to shut up and deal with it.
It used to be like this about the menopause too with women telling other women 'it's just natural' and get on with it. Thank god things are changing with the newer generation.
OP find out if your trust allows it. Some do. You can also write to the trust or the hospital outlining why you think you ought to get it done. - I personally wrote to my local dental hospital years ago and was offered full braces for free as an adult so you can get unexpected support.