TRACON Controller reported turning a aircraft off the ILS approach course to avoid a collision with an opposite direction aircraft not on their frequency resulting with the aircraft flying towards terrain in VMC conditions. The aircraft landed safely at the airport.

Narrative

I had Aircraft X on the ILS-Z Rwy XXR at ZZZ. I accepted a handoff from ZZZ tower on a Aircraft Y; climbing to 7500'. I went on the line and asked the tower to issue Aircraft Y to a 300 heading to keep him west of the localizer. There were frequency issues at the time. ZZZ1 normally uses XXX.X but there was a lot problems with that frequency; so I ended up using XXX.YY instead. About 5 minutes earlier; I told ZZZ tower about the frequency change. However; the tower controller forgot and switched the pilot to XXX.X instead.I noticed Aircraft Y was not turning and instead continuing northbound. I reached out to the tower again and said 'if the pilot comes back to you; can you put him on a 300 heading. I'm using XXX.YY' Then I hear on the shoutline; the tower controller said 'he came back to me; he has him in sight.' The tower controller NEVER issued the 300 heading. Aircraft Y continues to go right at Aircraft X. I got a conflict alert on my radar scope. Now; I feel there is an unsafe situation. So I tell Aircraft X 'offset to the left.' Aircraft X was below the MVA at this time. After they passed; Aircraft Y finally checks on the correct frequency.I told the supervisor at the time; what happened. They said they would look into it. 5 days later; the Supervisor confronts me and shows me the replay. The Supervisor then said 'they weren't even close.' However; they passed 1 mile apart laterally and I got a conflict alert. The Supervisor then criticizes my performance and told me I should of never told the pilot to 'offset to the left' because he was on an instrument approach. Instead; she tells me; if anything; I should have canceled his approach; tell him to continue inbound on the localizer; and climb him to 5000 (the MVA in the area). In my opinion; because the weather was great at the time; I figured a slight 'offset' was appropriate to use. Again; I was not talking to the Aircraft Y pilot and even though Aircraft Y pilot had Aircraft X in sight; he was pointed right at him. I took evasive action and did what I thought was safe and the most efficient.Recommendation: If ZZZ tower switched the pilot to the correct frequency initially; I could of vectored the pilot away from the localizer myself. But because there was confusion; I asked ZZZ tower to vector him for me. The tower controller did not comply with my instructions; even though it's my airspace and I have traffic in the way. Tower controllers and TRACON controllers see things differently. If I didn't tell Aircraft Y to offset; I'm pretty sure he would of responded to an RA creating a worse situation. However; I don't understand why the tower ignored my request. It's my airspace and as the receiving controller; I expected a 300 heading.