Aside from USAA (for eligible members and associate members) and GEICO, and for Visiting Forces and diplomats and a few other special cases, owning and insuring a UK car by a nonresident is difficult. Whether it is "legal" as such isn't the point: the question is whether there are practical workarounds. For one thing there are thousands of car owners in Britain without right of abode who keep cars at their vacation homes.
Here's one suggestion from a Daily Telegraph help column:
http://www.telegraph.co.uk/motoring/caradvice/honestjohn/7922584/How-you-can-insure-visitors-from-overseas-on-you-car.html There are lots of others to be found with a search engine.
And here's my own anecdotal experience: My daughter's ex-boyfriend, an Australian here on a visitor visa (he had the right to claim residence on the basis of his father's or grandfather's UK birth but this is beside the point as (1) he never did, and (2) neither DVLA nor his insurer knew that). At one point he returned to Australia, abandoning his car in the garage of my daughter's apartment building.
I rang DVLA and eventually worked out a viable solution, starting with an immediate SORN. I thought I could just get a car transporter (using "trade plates" - those red plates stuck onto cars driven by professional transport drivers and the subject of a 2013 Radio 4 programme:
http://www.bbc.co.uk/programmes/b021406q In the USA these are either transporter or dealer plates) but DVLA told me (I think wrongly) that trade plates can be used only by a firm having a financial interest in the car, so I couldn't get a "trade plater" to take the car to auction for me.
In the end this is what we did: There were no papers for the car. Nonetheless DVLA, based on the SORN and after a waiting period gave us some kind of form with which we sold the car to a used-car dealer for £400 and he took the car away with his trade plates. The former owner, in Australia, didn't mind.
On trade plate insurance, see this:
https://www.cglloyds.co.uk/commercial-insurance/motor-trade-insurance/trade-plate-insurance (not valid for "Social, Domestic and Pleasure"). Which reminds me of this, another possible (but temporary) workaround: if you bring a car to Dover with no plates at all (perhaps because you took them off while on the ferry to return them to the issuing government), you can buy Q plates from the AA at the port, get a Customs permit for (I think) 6 months, and buy insurance cover (any visitor can (and must) buy insurance cover for an imported car if s/he doesn't have a green-card or proof of UK insurance. (I have actually done this, but it was 30 years ago so procedures may have changed.) Recent TV footage of high-end Arabian-Gulf-registered cars being towed in Knightsbridge for lack of insurance says it all. (Once you have Q plates you can go to the AA's Basingstoke office and get a temporary international logbook. No wonder Britain is one of the world's favourite countries for car theft.)