From the photos (and thanx for posting your photos), it looks like it could be anterior open bite with bi-max recession of BOTH jaws. The upper jaw being behind the lower as seen by upper lip being behind lower lip would be deemed as 'class 3'. Maxilla (lefort 1 area) is clearly recessed evidenced by the conCAVE area right beside the nose. The chin is recessed and casts LONG in the frontal photo due to the 'hanging' soft tissue below. So, your chin bone might not be long but the lack of bone support to lower whole mandible area including chin results in what you see as what I call a 'sub-mental HANG' or GULLET formation where it's the BASE (bottom) of this 'sub-mental hang CURVE' you see from the FRONT which gives look of very long chin. The mandible recession is evidenced by the extremely short chin to neck distance.
The extreme aesthetic deviation from the norm, in NO WAY, justifies just 'fixing' the class 3 via ortho alone. On the contrary, the skeletal formation justifies BI-MAX surgery.
The facial appearance being one of extreme LACK of soft tissue support, to my eye, is indicative of bimax RETRUSION (retrusion of BOTH jaw bones). The class 3 look of it is from the maxilla being more retruded than the mandible BUT BOTH have significant retrusion.
Although you don't give a diagnosis of 'anterior open bite', (nor can I confirm one from here), the sub-mental HANG or 'GULLET' formation under chin is a common soft tissue contour associated with anterior open bite.
That said, my guess would be your surgery would POSSIBLY be posterior impaction; wedge shaped removal of back part of maxilla. Something where excess growth to BACK of maxilla (posterior maxilla) can thrust the lower jaw downward and back and contribute to that gullet formation. Removal of posterior excess helps relieve the downward thrust it causes to the mandible and also helps 'level out' the maxilla. From there, the maxilla is displaced forward and also the mandible and chin.
If you have SLEEP APNEA, and ALSO IF insurance is paying for the surgery, 'push' for surgical correction on THOSE grounds ALONE and don't tell them you are interested 'only in aesthetics'. Push for CONSULTS with maxfax surgeons for sleep apnea and for multiple consults. Once you get in for a consult, at THAT POINT, discuss aesthetic maximization possibilities with the surgery.
I don't know much about the NHS in the UK other than that they CAN and do FINANCE some maxfax surgeries. But when ever you are dealing with insurance BUREAUCRATS in any insurance or government system, there's always the chance (sometimes high probability) of getting an OFFICIOUS bureaucrat with whom the mention of 'aesthetic alone' motivation can TRIGGER them to SPIN THEIR GEARS to deny surgery on those grounds even if it's justified which I feel that it IS in your case.
IF insurance is NOT paying for the surgery and if it's SELF PAY, then perfectly appropriate to disclose from the get go that aesthetic improvement is your main focus and they will KNOW aesthetic improvement will ALSO come with correction to open up the airway.
