Where a customer has logged in and already has a shipping address recorded, it does not come up in the checkout - they have to fill it in again.
I just tried to replicate this issue on our demo store and the shipping address fills in. Is your website live somewhere I could test?
Please try here: https://web-staging.artlookonline.com/new-modernists-gallery/pages/64317/1438/woman-child-reading-by-the-fireside.
It could be related to the address being UK format? The address I have stored for shipping is 24 Park Street, Bristol, BS1 7XJ United Kingdom