We're currently in the middle of exactly this situation, not through choice, but because dc3 was halfway through UK NHS orthodontic treatment when he turned 18 and left for university in another country (in Europe, so logistically similar to your situ).
For us there's no question of fraud or non-entitlement, since dc is ordinarily resident in the UK, and the course of treatment (hospital based) was initiated while still in education in the UK. For the last two years when he's been out of the UK the logistics have been a massive, massive PITA (and significant cost), and that's with us having got the really hassly bits out of the way while dc was still in the UK.
Even if your dc qualified on clinical grounds, which are extremely tight - basically it needs to be medically essential, not just aesthetic - in order to initiate treatment in the UK you would have multiple assessment appointments here. The orthodontist or hospital would also want the details of your NHS GP and UK dentist, as all the letters are copied to the GP and dentist, so that would likely be a problem if you're not registered with a UK GP practice.
If the issues are serious enough to merit hospital treatment, that will involve at least three initial assessment appointments plus multiple x-rays - the x-ray appointments are often completely separate from any of the other appointments, and then there's another follow-up appointment to discuss the X-ray, and so on and on. Then once any surgery and follow-up have been completed and the brace installed, you have an appointment every six weeks or so to adjust the wires, and that's without allowing for any emergency broken brace appointments (particular frequent in the early days when the whole system is under maximum tension). It's unrealistic to think that an orthodontic practice in another country would take over the routine wire adjustments, as they won't want to be involved in an existing course of treatment which they have not initiated and for which they don't have the clinical oversight. You MAY be able to get a local orthodontist to do emergency broken brace repairs, but that would be at their discretion. Then will definitely not carry out routine adjustments, as it's not their treatment programme. We tried to get dc's care transferred to the country where he is studying, and no orthodontist was prepared to consider it, even with supporting documentation and letters from the clinicians here.
We live an hour's drive away from the hospital where dc is being treated, and that was bad enough when he was in the UK - since he's been abroad it has been a major headache. Bear in mind that routine appointments are often cancelled or rearranged at short notice - we have had that happen multiple times when dc had already booked and paid for a flight. Depending on how long the course of treatment takes, even if you are willing to take on board the logistical nightmare of travelling to the UK every few weeks, the costs of flights/cancelled flights/overnight stays would soon rack up to be equal or greater than the cost of local treatment, as it's almost impossible to align cheap flight dates with available appointments.
tl;dr: your plan is almost certainly not feasible, and even if by some miracle you managed to pull it off, you would regret your choice very quickly.