I'm not in a position to recommend any surgeons as I haven't had surgery yet, but just some general points: recessed jaw etc. is not really an illness that can be 'diagnosed' and then any doctor will just treat you on the basis of that diagnosis. It's your personal decision if you want to do something about your jaw being recessed, and every doctor you see will have his or her own idea about how to make your jaw / face look better, essentially. So in that sense, there's no point being 'diagnosed' in the UK if you are planning to have surgery elsewhere.
If you are a UK resident, there is a chance you qualify for free treatment under the NHS, depending on how 'bad' your situation is according to their guidelines. To find out if you qualify, the best way is probably to speak to an orthodontist first and they can refer you to a hospital. There are really long waiting lists, for example, most people wait one or two years just to start their orthodontic treatment (this is after they were told they qualify for free treatment - that process itself takes at least a few months). If you have to pay privately, it is very expensive, the starting price is about ten thousand pounds for single jaw surgery; most surgeons charge more, it's a lot more if it's double jaw surgery, and orthodontic treatment costs at least a few thousand pounds on top of that.
The other thing is, if you decide to go for jaw surgery instead of just genioplasty or implants etc., you will need braces and the surgeon and orthodontist will have to work closely together. Most surgeons will want you to go to a specific orthodontist that they usually work with. So if you want surgery outside the UK, you will need to find an orthodontist in the UK that agrees to work with a random surgeon in another country and that might be difficult but hopefully not impossible.
