ATC Tower Controller reported working an emergency aircraft not on frequency and asking the approach control to clear the final approach course which was not accomplished resulting in a dangerous situation with the Tower not talking to the emergency aircraft and having other traffic inbound to the airport.

Narrative

Aircraft X departed ZZZ1 IFR to ZZZ area. ZZZ1 [Tower] called ZZZ to advise aircraft was unable to control instruments and heading towards ZZZ. I then made a call to the approach control (R scope) to advise of emergency inbound aircraft that was unable to control altitude or control of his plane and to break off all guys on final because we were unsure of which runway they were trying to land. ZZZ [TRACON] did not break off aircraft on final. ZZZ was not in talks with the aircraft until they touched down RWY XXL and spoke to ground control. Aircraft X did a go around due to being too fast; he was lined up for RWY XXR but switched to the north side. ZZZ did everything to protect aircraft landing while ZZZ [TRACON] was worried about not breaking off traffic inbound. They also adempted to vector Aircraft X behind Aircraft Y vs proving services to the emergency who had stated several times he was not in control of his plane and needed to land.Recommendation: ZZZ [TRACON] handle emergency aircraft as top priority and be willingly to resequence aircraft even if it adds to their workload.