I’m assuming that your loft is the English equivalent to our attics? (Over here, a loft is a type of apartment, usually built in old warehouses and other structures that had a previous life as something else.)
You also have a loft in a barn when the hay used to stored. Also we have apartments (flats) as you mention normally called loft or attic flats or apartments.
Last year I fence blew down and our neighbours put a new one up for us (we of course paid for the materials) on the left you can see our pigeon loft. As loft can also be a building in a garden for pigeons we have a pigeon loft in our garden which is really just a large shed designed especially for pigeons.