"they somehow register it as a US a/c, service it as a US a/c but fly it with UK crews"
When the purchase decision was first announced, that is exactly how some of the newspapers described the deal. The implication was that the UK was simply funding three extra airframes for a joint pool.
Now I can't remember where I read that, but at the time the impression was clear. These would be UK financed USA aircraft with UK crews - according to those press reports