What is the economic strategy of the UK? The UK has run a current account deficit running into billions of GBP per annum since the turn of the century.

If we apply the mindset of a household to the economy then clearly this would appear to be unsustainable. But presumably the household mindset does not exactly map onto a country's economy.

What macroeconomic strategy is this and why can the UK get away with this (or can't it)?
You are correct in saying that a household is not comparable to the U.K. or any other country. A household does not issue currency and bonds and typically cannot issue debt obligations that are good beyond, say, approximately 30 years. A person in a household can borrow for approximately 30 years to buy real estate but it would be difficult for that person to borrow for 100 years. Contrast this to the U.K. government which can easily borrow for 30 years and even beyond 100 years. I am reasonably sure the U.K. will collect taxes from people that haven't yet been born and it will collect taxes from companies that have not yet been founded so the value of a long maturity bond appears to be supportable. Likewise some existing U.K. companies and some that will exist in the future can sell long dated bonds.  Existing U.K. companies and those that will exist in the future will also be able to sell shares in their equity and the buyer usually has no right to sell them back to the company, ever. That could mean a long time. If the equity being sold to foreigners is land then that is sustainable only until the supply of land on the islands of the U.K. has been exhausted.

The money raised by Britons working to produce exports pays for most of the imports. Additional to that, the money raised by the U.K. government and U.K. companies selling bonds and shares can be used to buy imports as long as outsiders want to buy bonds and shares. That might be many years so it might be sustainable for a long time. I don't know if this is as much a directed or planned strategy as it is a result that spontaneously emerges from a multitude of foreigners simply wanting to own U.K. real estate and bonds and shares and the U.K. willing to sell those assets. Additional to that, the U.K. receives interest, principal repayment, and dividends from abroad. The sustainability of these depends upon the particulars.