There is a button on the side to adjust up or down, but how can the headrest be removed completely so I can fit a coat hanger to the shafts?

Have you thought of something like this one?

https://www.amazon.co.uk/Car-Headrest-Hanger-Luxury-Detachable/dp/B01H5QI6GG/ref=sr_1_22?ie=UTF8&qid=1468177441&sr=8-22&keywords=headrest+coat+hanger

I have an older one of these in mine (came from BMW at silly money) and it has a spring that pushes the ends into place against the columns. No need to remove anything.

There is a button on the side to adjust up or down, but how can the headrest be removed completely so I can fit a coat hanger to the shafts?

You need to push down some of the leather/cloth on the seat back top where the button is to control the headrest. If you can, you'll reveal a small opening within the mechanism. Push a flat headed screwdriver into it and pull the headrest up and it should slide out entirely.

Of course, drop the seat as far back as you can to make this easier as the headrest won't come out if the seats upright.

A bit fiddly, but it works. And I managed to fit my old Skoda wooden coat hander into mine which looks great with the Beige leather and is used everyday.

No need for a screwdriver, the slot is shaped specifically for the main car key to fit.