Hi

You could always cover it in horticultural fleece to keep it snug

You are right to worry about keeping it in a heated environment . Your balcony is extremely sheltered - it has a ceiling which keeps off the worst of the weather, and you're in London, your Olive tree will be fine outdoors. It might, though need a larger pot, I suspect its rootbound - given the weather at the moment, if it is rootbound, its probably wise to insulate the roots, so if you can get hold of a cardboard box, stand the pot inside and pack round with shredded newspaper, anything that insulates, until you're able to repot it into something larger. And, as Pam above says, you can drape horticultural fleece over the top if you're really worried - but don't leave that in place all winter, only while its very, very cold.

Bear in mind that pots only freeze solid if the weather remains below zero deg C night AND day, which is pretty unusual in the UK, especially where you are.

Bamboo, Gnarly,
Thanks for your advice. I think I'll leave it outside but maybe bring it closer to the inner part of or the balcony so it has more shelter. I'm a bit hesitant to use a cardboard box with shredded newspaper because it will just soak through and rot. I feel so sorry for the little guy standing out there on his own. If he survives the winter I'll be a miracle. These are harsh conditions for warm weather plants to deal with.

I only meant use the box and paper as a temporary measure whilst it is so cold, otherwise use some bubblewrap, because you're right, it is pretty cold. Moving it closer to your apartment is a good idea, but move it back out when the weather becomes mild again.

If its any comfort, I've got two Pelargoniums (geraniums) and a tender trailing fuchsia still in full flower on my windowsill outside on my own balcony, despite all this cold weather - they're partially covered by a roof that doesn't extend as far as yours, and being on the windowsill, benefit from being close to the interior.