Would it be possible to replace the rivets with tiny screws and nuts ?

Hard to to give exact advise without knowing what they look like, but it should be an easy enough job to drill out the original rivets and replace them with new ones. Depending on the type of rivet that was originally used threre are many options available. I'd be surprised if the originals are POP-style rivets, and would guess that cap rivets were used, which is good because they are available in many sizes and materials such as copper, nickle, and chrome, so I'm sure you should be able to come very close to matching the originals. 

Photo? You just need to measure an existing rivet and see how close you can come with a replacement. Most important will be the thickness of material that the rivet has to hold together, along with the diameter of its shank.