B737-800
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Communication Breakdown
ATC Issue All Types; Conflict NMAC
Flight Crew Executed Go Around / Missed Approach
Airspace Structure; Procedure
Responded to an RA on TCAS at 4;600 MSL. Advised ATC; ZZZ Approach Control. No response; 3 times we told Approach responding to RA. No response. Executed GA climbed out on LOC to 6;000 feet then they gave us Fly LOC to 6;000 feet. They did not respond to our 'Responding to RA' calls. I (Pilot Flying) was not actively monitoring guard 121.5; by my mistake I did not have it selected. Our radios did appear there was some static; but not a factor. We heard all other aircraft; I'm guessing ATC was shift change? Or ATC was saturated. ZZZ is a saturated airport. They normally stagger [Runway] XXR aircraft on approach with those on [Runway] XXC. Our RA was virtually next to us and within 400 feet of our altitude.
Air carrier Captain reported receiving a TCAS RA and having difficulty notifying ATC of the resulting clearance deviation. Crew executed a missed approach.