How to update view from callback inside of custom delegate class?

Multiple issues with your code.

1. You are initializing GlobalVarsModel twice. Once in the View and once in the delegate. So changes in one won´t reflect in the other.

2. You are implementing the delegate in a subclass of your AVSpeechSynthesizer therefor it is capsulated in it and you can´t update your View when an event arises.

I changed the implementation to address this issues:

class GlobalVarsViewmodel: NSObject, ObservableObject { //You need to derive from NSObject first, because `AVSpeechSynthesizer` is `objc` related
    @Published var prayerAudioID: UUID?
    @Published var uttPrayerAudio = ""
    @Published var strAudioBtnImgStr = "play.fill"
    @Published var strAudioBtnText = "Play Audio"
    let audioSession = AVAudioSession.sharedInstance()
    var synthesizer = CustomAVSpeechSynth()
    
    override init(){
        super.init()
        synthesizer.delegate = self // assign the delegate
    }
}

extension GlobalVarsViewmodel: AVSpeechSynthesizerDelegate{ // extend the viewmodel to implement the delegate
    
    func speechSynthesizer(_ synthesizer: AVSpeechSynthesizer, didPause utterance: AVSpeechUtterance) {
    }
    
    func speechSynthesizer(_ synthesizer: AVSpeechSynthesizer, didStart utterance: AVSpeechUtterance) {
    }
    
    func speechSynthesizer(_ synthesizer: AVSpeechSynthesizer, didCancel utterance: AVSpeechUtterance) {
    }
    
    func speechSynthesizer(_ synthesizer: AVSpeechSynthesizer, didFinish utterance: AVSpeechUtterance) {
        print("Finished praying.")
        strAudioBtnImgStr = "play.fill" // here assign the text and button appearance
        strAudioBtnText = "Play Audio"
    }
    
    func speechSynthesizer(_ synthesizer: AVSpeechSynthesizer, didContinue utterance: AVSpeechUtterance) {
    }
    
    func speechSynthesizer(_ synthesizer: AVSpeechSynthesizer, willSpeakRangeOfSpeechString characterRange: NSRange, utterance: AVSpeechUtterance) {
    }
}
// I don´t think you need this anymore
class CustomAVSpeechSynth: AVSpeechSynthesizer {
    
    //NOT DESIRED OUTPUT LIST
    //@Published
    //@ObservedObject
    //@State
}

struct TappedPrayerView: View {
    var tappedPrayer: Prayer
    @StateObject private var gVars = GlobalVarsViewmodel()
    @Environment(\.scenePhase) var scenePhase
    
    var body: some View {
        ScrollView {
            VStack {
                Text(tappedPrayer.strTitle).font(.title2).padding()
                HStack {
                    Spacer()
                    Button {
                        gVars.prayerAudioID = tappedPrayer.id
                        gVars.uttPrayerAudio = tappedPrayer.strText
                        
                        if (gVars.strAudioBtnText == "Play Audio") {
                            gVars.strAudioBtnImgStr = "pause.fill"
                            gVars.strAudioBtnText = "Pause Audio"
                            if (gVars.synthesizer.isSpeaking || gVars.synthesizer.isPaused) {
                                gVars.synthesizer.stopSpeaking(at: .immediate)
                                gVars.synthesizer.speak(AVSpeechUtterance(string: gVars.uttPrayerAudio))
                            } else {
                                gVars.synthesizer.speak(AVSpeechUtterance(string: gVars.uttPrayerAudio))
                            }
                        } else if (gVars.strAudioBtnText == "Pause Audio") {
                            gVars.synthesizer.pauseSpeaking(at: .immediate)
                            gVars.strAudioBtnImgStr = "play.fill"
                            gVars.strAudioBtnText = "Continue Audio"
                        } else if (gVars.strAudioBtnText == "Continue Audio") {
                            if (gVars.synthesizer.isPaused) {
                                gVars.synthesizer.continueSpeaking()
                                gVars.strAudioBtnImgStr = "pause.fill"
                                gVars.strAudioBtnText = "Pause Audio"
                            }
                        }
                    } label: {
                        Label(gVars.strAudioBtnText, systemImage: gVars.strAudioBtnImgStr).font(.title3).padding()
                    }.onAppear {
                        if ((gVars.synthesizer.isSpeaking || gVars.synthesizer.isPaused) && tappedPrayer.id != gVars.prayerAudioID) {
                            gVars.strAudioBtnImgStr = "play.fill"
                            gVars.strAudioBtnText = "Play Audio"
                        }
                    }
                    Spacer()
                    Button {
                        if (gVars.synthesizer.isSpeaking || gVars.synthesizer.isPaused) {
                            gVars.synthesizer.stopSpeaking(at: .immediate)
                            gVars.strAudioBtnImgStr = "play.fill"
                            gVars.strAudioBtnText = "Play Audio"
                            gVars.prayerAudioID = UUID(uuidString: String(Int.random(in: 0..<7)) + (gVars.prayerAudioID?.uuidString ?? "777"))
                        }
                    } label: {
                        Label("Restart", systemImage: "restart.circle.fill").font(.title3).padding()
                    }
                    Spacer()
                }
                Spacer()
                Text(tappedPrayer.strText).padding()
                Spacer()
            }
        }.onAppear {
            if (gVars.synthesizer.isPaused) {
                if (tappedPrayer.id == gVars.prayerAudioID) {
                    gVars.strAudioBtnImgStr = "play.fill"
                    gVars.strAudioBtnText = "Continue Audio"
                }
            } else if (gVars.synthesizer.isSpeaking) {
                if (tappedPrayer.id == gVars.prayerAudioID) {
                    gVars.strAudioBtnImgStr = "pause.fill"
                    gVars.strAudioBtnText = "Pause Audio"
                }
            } else {
                gVars.strAudioBtnImgStr = "play.fill"
                gVars.strAudioBtnText = "Play Audio"
            }
        }.onChange(of: scenePhase) { newPhase in
            if (newPhase == .active) {
            } else if (newPhase == .inactive) {
            } else if (newPhase == .background) {
            }
        }
    }
    
    struct TappedPrayerView_Previews: PreviewProvider {
        static var previews: some View {
            let defaultPrayer = Prayer(strTitle: "Default title", strText: "Default text")
            TappedPrayerView(tappedPrayer: defaultPrayer)
        }
    }
}

Remarks:

• I changed the name of GlobalVarsModel to GlobalVarsViewmodel because it is exactly that, a Viewmodel.
• I changed the synthesizer var to be instance related instead of static
• same for the AVAudioSession

Edit to adress the comment for clarification: I changed the implementation from static because it is not needed here. You can read more about it here -> https://www.donnywals.com/effectively-using-static-and-class-methods-and-properties/