Air carrier FO reported difficulty contacting a busy Approach Controller. This resulted in a missed approach to avoid an NMAC with another air carrier.

Narrative

As we began to descend via the ZZZZZ into ZZZ approach began to slow us down as it appeared we were following traffic ahead of us. Once we arrived at ZZZZZ we were given a vector of a heading 250 and an altitude of 6;500. This heading was going to establish us on the final approach course for the ILS XXR. The airport was VFR and the approach controller was very busy on his frequency. It appeared he was managing both arrivals into XXL and XXR. Once we cleared terrain the captain and I began to grow concern as we were just outside of ZZZZZ1 (fix prior to FAF) at 6;500 and not given any lower altitudes or clearance to join the localizer. The radios were so busy I couldn't even chime in to ask for lower without stepping on someone. The controller eventually gave us lower and clearance to intercept (1;000 ft above FAF altitude) but in hindsight we were going to be unstable anyways. We got dirty as quick as we could dropped the gear; spoilers and flaps in an attempt to regain stability. While this is happening Aircraft Y hops on frequency and warns the approach controller that they are overshooting their final approach course on XXL. Although I heard this I'm really dialed in on backing up the pilot flying and monitoring the aircraft is this critical phase of flight. But the moment I glance outside sure enough what feels like 500 feet below us (due to our steep decent angle) Aircraft Y is feet away from our final approach course veering back onto there final approach course. Immediately I hop on the radios stating that I have visual of the 737 right below and in front of us. ATC failed to warn us of the aircraft; vector us off course or give us a higher altitude until I hopped on the radios. Once I advised ATC they insisted we level off at 3;500 but at this point we are coming up on the FAF; task saturated; and in shock of how close we were from another aircraft that we immediately went around. The go-around was a non event however they still kept us low and headed towards terrain with the radios still being majorly congested. We queue in on the radios asking ATC what was the plan because all along it felt as if they were continuously forgetting about us. After advocating for ourselves we got vectors back onto course; and after advocating once again (seconds from flying over the localizer inbound) we were cleared for the visual approach into XXR. We asked for a number once we were on the ground hoping for some form of justification but were only given that they in fact kept us high; and Aircraft Y was the aircraft in the wrong and that they were training as well.