dat581 is partly correct except our missiles were bought and paid for.
What confuses many people is the King's Bay amendment to the Polaris Sales Agreement (yes, it's still called that). The UK missiles were originally to be maintained in RNAD Coulport, almost nextdoor to Faslane. The US Government proposed that there would be a financial advantage in processing our missiles through their existing facility in King's Bay, GA. Part of the agreement was that missiles passing through Kings Bay would be "pooled"; the missile we put in would not necessarily be the same missile we receive back.
The missiles sat quietly in the back of a V Boat are most certainly British property.