下図のように編集開始時にDatePickerを起動させ，時間変更時に変更時間をTextFieldに反映させます．

まずTextFieldのDelegateを追加します．

ViewController.swift

class ViewController: UIViewController, UITextFieldDelegate

次に以下のコードを追加hします

ViewController.swift


var toolBar:UIToolbar!
@IBOutlet var wakeTimeTextField: UITextField!

override func viewDidLoad() {
    super.viewDidLoad()
        
    wakeTimeTextField.delegate = self
    setupToolbar()
}
func setupToolbar() {
        //datepicker上のtoolbarのdoneボタン
        toolBar = UIToolbar()
        toolBar.sizeToFit()
        let toolBarBtn = UIBarButtonItem(title: "DONE", style: .plain, target: self, action: #selector(doneBtn))
        toolBar.items = [toolBarBtn]
        wakeTimeTextField.inputAccessoryView = toolBar
}
    
func textFieldDidBeginEditing(_ textField: UITextField) {
    let datePickerView:UIDatePicker = UIDatePicker()
    datePickerView.datePickerMode = UIDatePicker.Mode.time
    textField.inputView = datePickerView
    datePickerView.addTarget(self, action: #selector(datePickerValueChanged(sender:)), for: UIControl.Event.valueChanged)
}
    
    //datepickerが選択されたらtextfieldに表示
@objc func datePickerValueChanged(sender:UIDatePicker) {
    let dateFormatter = DateFormatter()
    dateFormatter.dateFormat  = "H:mm";
    wakeTimeTextField.text = dateFormatter.string(from: sender.date)
}

//toolbarのdoneボタン
@objc func doneBtn(){
    wakeTimeTextField.resignFirstResponder()
}