Can my upturned nose be fixed so I look less piggy? (photo)

The revision rhinoplasty for this situation is challenging, it will involve the lengthening of the nose to reduce the upturned appearance and will require grafting of cartilage. The NHS system will certainly not cover. You have no choice but to use a private surgeon either in UK or abroad. RegardsDr JDisclaimer:
This answer is not intended to give a medical opinion and does not substitute
for medical advice. The information presented in this posting is for patients’
education only. As always, I encourage you to see your personal physician for
further evaluation of your individual case.

I agree that you will need cartilage grafts to help push down the tip of your nose.  If you had previous breathing issues it is likely that the septal cartilage has been harvested and you may end up requiring a rib cartilage graft to push the tip downward.  If there is no breathing issue it is unlikely that there will be any sort of insurance coverage for the problem. 

Yes this can be helped with a further procedure.  I write from Canada, and like the NHS, the surgery is for appearance and is not covered under public health insurance.

Cartilage grafts will be necessary to lengthen your nose. I doubt any insurance company, either private or sponsored by the government, will pay for this cosmetic procedure. Have the work is done by a nasal specialist; you want to avoid making it worse.