How to preserve space occupied by status bar when hiding status bar animately?

You can use additionalSafeAreaInsets to add a placeholder height, substituting the status bar.

But for devices with a notch like the iPhone 12, the space is automatically preserved, so you don't need to add any additional height.

class ViewController: UIViewController {

    var statusBarHidden: Bool = false /// no more computed property, otherwise reading safe area would be too late
    override var prefersStatusBarHidden: Bool {
        return statusBarHidden
    }

    override var preferredStatusBarUpdateAnimation: UIStatusBarAnimation{
        return .slide
    }
    
    @IBAction func showButtonPressed(_ sender: Any) {
        statusBarHidden.toggle()
        if statusBarHidden {
            sideMenuWillAppear()
        } else {
            sideMenuWillDisappear()
        }
    }
    
    lazy var overlayViewController: UIViewController = {
        let storyboard = UIStoryboard(name: "Main", bundle: nil)
        return storyboard.instantiateViewController(withIdentifier: "OverlayViewController")
    }()
    
    var additionalHeight: CGFloat {
        if view.window?.safeAreaInsets.top ?? 0 > 20 { /// is iPhone X or other device with notch
            return 0 /// add 0 height
        } else {
            /// the height of the status bar
            return view.window?.windowScene?.statusBarManager?.statusBarFrame.height ?? 0.0
        }
    }
}

extension ViewController {
    
    /// add placeholder height to substitute status bar
    func addAdditionalHeight(_ add: Bool) {
        if add {
            if let navigationController = self.navigationController {
                /// set insets of navigation controller if you're using navigation controller
                navigationController.additionalSafeAreaInsets.top = additionalHeight
            } else {
                /// set insets of self if not using navigation controller
                self.additionalSafeAreaInsets.top = additionalHeight
            }
        } else {
            if let navigationController = self.navigationController {
                /// set insets of navigation controller if you're using navigation controller
                navigationController.additionalSafeAreaInsets.top = 0
            } else {
                /// set insets of self if not using navigation controller
                self.additionalSafeAreaInsets.top = 0
            }
        }
    }
    
    func sideMenuWillAppear() {
        
        addChild(overlayViewController)
        view.addSubview(overlayViewController.view)
        overlayViewController.view.frame = view.bounds
        overlayViewController.view.frame.origin.x = -400
        overlayViewController.view.autoresizingMask = [.flexibleWidth, .flexibleHeight]
        overlayViewController.didMove(toParent: self)
        
        addAdditionalHeight(true) /// add placeholder height
        
        UIView.animate(withDuration: 1) {
            self.overlayViewController.view.frame.origin.x = -100
            self.setNeedsStatusBarAppearanceUpdate() /// hide status bar
        }
    }

    func sideMenuDidAppear() {}

    func sideMenuWillDisappear() {
        
        addAdditionalHeight(false) /// remove placeholder height
        
        UIView.animate(withDuration: 1) {
            self.overlayViewController.view.frame.origin.x = -400
            self.setNeedsStatusBarAppearanceUpdate() /// show status bar
        } completion: { _ in
            self.overlayViewController.willMove(toParent: nil)
            self.overlayViewController.view.removeFromSuperview()
            self.overlayViewController.removeFromParent()
        }
    }

    func sideMenuDidDisappear() {}
}

Result (Tested on iPhone 12, iPhone 8, iPad Pro 4th gen):

iPhone 12 (notch)	iPhone 8 (no notch)

iPhone 12 + navigation bar	iPhone 8 + navigation bar

Demo GitHub repo

First of all, it is not currently possible to make UINavigationController behave this way. However you can wrap your UINavigationController instance in a Container View Controller.

This will give you control over managing the top space from where the UINavigationController view layout starts. Inside this container class, you could manage it like following -

class ContainerViewController: UIViewController {

    private lazy var statusBarBackgroundView: UIView = {
        let view = UIView(frame: .zero)
        view.backgroundColor = .clear
        view.translatesAutoresizingMaskIntoConstraints = false
        return view
    }()

    private lazy var statusBarBackgroundViewHeightConstraint: NSLayoutConstraint = {
        statusBarBackgroundView.heightAnchor.constraint(equalToConstant: 0)
    }()
    
    var statusBarHeight: CGFloat {
        if #available(iOS 13.0, *) {
            guard let statusBarMananger = self.view.window?.windowScene?.statusBarManager 
            else { return 0 }
            return statusBarMananger.statusBarFrame.height
        } else {
            return UIApplication.shared.statusBarFrame.height
        }
    }

    var statusBarHidden: Bool = false {
        didSet {
            self.statusBarBackgroundViewHeightConstraint.constant = self.statusBarHidden ? self.lastKnownStatusBarHeight : 0
            self.view.layoutIfNeeded()
        }
    }

    private var lastKnownStatusBarHeight: CGFloat = 0

    override func viewDidLoad() {
        super.viewDidLoad()
        
        let topView = self.statusBarBackgroundView
        self.view.addSubview(topView)
        NSLayoutConstraint.activate([
            topView.topAnchor.constraint(equalTo: self.view.topAnchor),
            topView.leadingAnchor.constraint(equalTo: self.view.leadingAnchor),
            statusBarBackgroundViewHeightConstraint,
            topView.trailingAnchor.constraint(equalTo: self.view.trailingAnchor),
        ])
    }
    
    override func viewDidLayoutSubviews() {
        super.viewDidLayoutSubviews()
        
        let height = self.statusBarHeight
        if height > 0 {
            self.lastKnownStatusBarHeight = height
        }
    }

    func setUpNavigationController(_ navCtrl: UINavigationController) {
        self.addChild(navCtrl)
        navCtrl.didMove(toParent: self)
        self.view.addSubview(navCtrl.view)
    
        navCtrl.view.translatesAutoresizingMaskIntoConstraints = false
        NSLayoutConstraint.activate([
            navCtrl.view.topAnchor.constraint(equalTo: statusBarBackgroundView.bottomAnchor),
            navCtrl.view.leadingAnchor.constraint(equalTo: self.view.leadingAnchor),
            navCtrl.view.bottomAnchor.constraint(equalTo: self.view.bottomAnchor),
            navCtrl.view.trailingAnchor.constraint(equalTo: self.view.trailingAnchor),
        ])
    
        self.view.layoutIfNeeded()
    }

}

Now from your call site, you can do following -

class ViewController: UIViewController {
    
    var statusBarHidden: Bool = false {
        didSet {
            UIView.animate(withDuration: Constants.config_shortAnimTime) { () -> Void in
                
                /// Forward the call to ContainerViewController to act on this update
                (self.navigationController?.parent as? ContainerViewController)?.statusBarHidden = self.statusBarHidden
                
                /// Keep doing whatever you are doing now
                self.setNeedsStatusBarAppearanceUpdate()
            }
        }
    }

}