I would think she would be classed as single. You are not living together permanently as a couple.

It is the address where the car is registered as being kept that is important.

If the car is in your name, then the Insurance should be in your name, with her as a named driver. When she passes her test, be prepared for a large increase in premium. The Insurers charge a reduced premium for provisional license holders, as they are supervised when driving. Once they can drive on their own as an inexperienced driver, the premium will shoot up. If she then takes the car with her to London, she will become the main driver and the car/Insurance, will need to be registered with her in London.