Low or no service after screen repair

These are difficult phones to work on so I can see how this could happen. Sony phones require a special sealing procedure to prevent the screen and back cover from separating. I would recommend looking for a shop that specializes in Sony repairs or bring it back to the shop you had it repaired at. You may still have warranty through them. Ifixit has a large list of reputable shops also. As far as the cellular connectivity issue goes, there are many small contacts and antennas inside the xperia's that can be interrupted . I would recommend having a professional check all these connections and replace the main antenna board while he/she has the phone opened up.