Developed a lump after breaking my nose at the bridge. What could it be?

Swelling, particularly at the bridge of the nose, is a common result of nasal trauma, whether or not there is an actual fracture.  Initially, it is important to use ice and elevation, just like in any other broken or bruised bone.  Early swelling that can be compressed, as in your case, will at least partially resolve.  However, pressure taping will help it resolve more quickly and more completely.  If the swelling persists and becomes firm, scar tissue has developed.  A small injection of steroids should help some.  Frequently, however, this scar tissue becomes permanent and requires surgical correction.

When the nasal bones are broken there can be bleeding and swelling that lift the layer on top of the bone. This space is probably what moves and goes away when you press on your nose. New bone can form in this space and form a permanent bump. Injection of steroids in the space may decrease the swelling and decrease new bone formation. 

Of course, this is difficult to know without photos and exam, but because you can flatten it and because it returns slowly it may just be inflammation/swelling. It could be a small pice of bone/cartilage though, in my opinion, lass likely so. At any rate, an initial exam by a plastic surgeon possibly followed by another exam a few weeks/months later should help identify the cause and the treatment options----IF it persists.