The best way to react on application changes while using the MVVM pattern would be to implement a IThemeService interface as shown in your link.

xamarin forms iOS

But I think it's not possible to react to configuration changes in Xamarin.Forms.iOS plattform while using MvvmCross. I looked into the source code of the MvvmCross.Forms.iOS project and couldn't find any equivalent to the MvvmCross.Forms.Android setup methods like OnConfigurationChanged.

On Android you can easily refresh the app-theme while change the system theme in the MainActivity.

    public class MainActivity : MvxFormsAppCompatActivity
    {
        public override void OnConfigurationChanged(Configuration newConfig)
        {
            base.OnConfigurationChanged(newConfig);
            this.UpdateTheme(newConfig);
        }

        protected override void OnResume()
        {
            base.OnResume();
            UpdateTheme(Resources.Configuration);
        }

        protected override void OnStart()
        {
            base.OnStart();
            this.UpdateTheme(Resources.Configuration);
        }

        private void UpdateTheme(Configuration newConfig)
        {
            if (Build.VERSION.SdkInt >= BuildVersionCodes.Froyo)
            {
                var uiModeFlags = newConfig.UiMode & UiMode.NightMask;
                switch (uiModeFlags)
                {
                    case UiMode.NightYes:
                        Mvx.IoCProvider.Resolve<IThemeService>().UpdateTheme(BaseTheme.Dark);
                        break;

                    case UiMode.NightNo:
                        Mvx.IoCProvider.Resolve<IThemeService>().UpdateTheme(BaseTheme.Light);
                        break;

                    default:
                        throw new NotSupportedException($"UiMode {uiModeFlags} not supported");
                }
            }
        }
    }

But in the AppDelegate on the iOS plattform, you don't have any of these functionalitys to override.

    public class AppDelegate : MvxFormsApplicationDelegate
    {
        public override bool FinishedLaunching(UIApplication application, NSDictionary launchOptions)
        {
            return base.FinishedLaunching(application, launchOptions);
        }
    }

I copied this code from this project.

native xamarin iOS

When you are using native iOS you could override the TraitCollectionDidChange method. It's the equivalent to the android OnConfigurationChanged function. Maybee look here for more details. I adapted the android version to iOS for you. At First, you have to create a custom view controller.

// your supported theme versions
 public enum BaseTheme
    {
        Inherit = 0,
        Light = 1,
        Dark = 2
    }

public class MyViewController : UIViewController
    {
        public override void TraitCollectionDidChange(UITraitCollection previousTraitCollection)
        {
            base.TraitCollectionDidChange(previousTraitCollection);

            if (TraitCollection.UserInterfaceStyle != previousTraitCollection.UserInterfaceStyle)
            {
                UpdateTheme(TraitCollection.UserInterfaceStyle);
            }
               
        }

        private void UpdateTheme(UIUserInterfaceStyle newConfig)
        {
            switch(newConfig)
            {
                case UIUserInterfaceStyle.Dark:
                    Mvx.IoCProvider.Resolve<IThemeService>().UpdateTheme(BaseTheme.Dark);
                    break;

                case UIUserInterfaceStyle.Light:
                    Mvx.IoCProvider.Resolve<IThemeService>().UpdateTheme(BaseTheme.Light);
                    break;

                default:
                    throw new NotSupportedException($"UiMode {uiModeFlags} not supported");
            }      
        }
    }

I uploaded a project where I simplify coded an implementation for native IOS and android here. Complete and improve some things and it will work. Also look at the StarWars and TipCalc Project in the mvvmcross sample repo.

mvvmcross ioc

your interface structure could look like so; IThemeService (base project) - ThemeService (base project) - ThemeService(iOS project)

And you have to register the interface of course.

Mvx.IoCProvider.RegisterSingleton<IThemeService>(() => new ThemeService());