Your relative should have no problem buying a car but (as I understand it) the address on the V5 logbook needs to be an address where your relative can be contacted if necessary (e.g. if you get a speeding fine or parking ticket). Lots of people don't have a fixed address so this can be a friend or family member's address used as a forwarding address.

Insurance will be a little more expensive than a UK resident with a UK driving licence so it's worth shopping around for quotes. I'm sure they'd bump up the premiums for a driver with a six-month policy, too, but it'd still be possible to get insurance.

A Honda Civic would be a pretty sensible option for £6000 – look for a 2012-onwards model and it'll be reliable/comfortable enough for touring Europe.