3 текстовых поля, 1 вид выбора, но при использовании pickerView выбор не отображается в текстовом поле

Я думаю, что ваш код неверен в didSelectRow. проверьте первый и второй if, оба работают в одном и том же состоянии.

if teamNameField.isFirstResponder {
   let itemselected = hosp[row]
   hospNameField.text = itemselected
} else if teamNameField.isFirstResponder {
   let itemselected = team[row]
   teamNameField.text = itemselected
} else if mdNameField.isFirstResponder {
   let itemselected = mds[row]
   mdNameField.text = itemselected
}

Я думаю, вам следует изменить первое if на if hospNameField.isFirstResponder {..}

You have forgotten to subscribe the delegate and data sources to the PickerView..

import UIKit

class ViewController: UIViewController, UIPickerViewDataSource, UIPickerViewDelegate, UITextFieldDelegate {

    @IBOutlet weak var hospNameField: UITextField!
    @IBOutlet weak var teamNameField: UITextField!
    @IBOutlet weak var mdNameField: UITextField!
    @IBOutlet weak var selectionsPickerView: UIPickerView!

    var hosp = ["hosp1", "hosp2", "hosp3"]
    var team = ["team1", "team2", "team3"]
    var mds = ["MD1", "MD2", "MD3"]

    var itemSelected = ""
    var textFieldSelected = UITextField()

    weak var pickerView: UIPickerView?

    override func viewDidLoad() {
        super.viewDidLoad()
        // Do any additional setup after loading the view.

        let pickerView = UIPickerView()
        pickerView.delegate = self
        pickerView.dataSource = self

        hospNameField.inputView = pickerView
        teamNameField.inputView = pickerView
        mdNameField.inputView = pickerView

        hospNameField.delegate = self
        teamNameField.delegate = self
        mdNameField.delegate = self

        self.pickerView = pickerView
        /**
         the below lines are mandatory for

         pickerView(_ pickerView: UIPickerView, didSelectRow row: Int, inComponent component: Int)

         to execute
         */
        self.pickerView?.delegate = self
        self.pickerView?.dataSource = self
    }

    func numberOfComponents(in pickerView: UIPickerView) -> Int {
        return 1
    }

    func textFieldDidBeginEditing(_ textField: UITextField) {
        textFieldSelected = textField
    }
    // add this delegate and reload here
  func textFieldShouldBeginEditing(_ textField: UITextField) ->
        Bool {
            self.pickerView?.reloadAllComponents()
            return true
    }

    func pickerView(_ pickerView: UIPickerView, numberOfRowsInComponent component: Int) -> Int {
        if hospNameField.isFirstResponder {
            return hosp.count
        } else if teamNameField.isFirstResponder {
            return team.count
        } else if mdNameField.isFirstResponder {
            return mds.count
        }
        return 0
    }

    func pickerView(_ pickerView: UIPickerView, titleForRow row: Int, forComponent component: Int) -> String? {
        if hospNameField.isFirstResponder {
            return hosp[row]
        } else if teamNameField.isFirstResponder {
            return team[row]
        } else if mdNameField.isFirstResponder {
            return mds[row]
        }
        return nil
    }

    func pickerView(_ pickerView: UIPickerView, didSelectRow row: Int, inComponent component: Int) {
        /**
         type here, need to check the responder for hospNameField also
         you have executed teamNameField.isFirstResponder twice
         */
        if hospNameField.isFirstResponder {
            let itemselected = hosp[row]
            hospNameField.text = itemselected
        } else if teamNameField.isFirstResponder {
            let itemselected = team[row]
            teamNameField.text = itemselected
        } else if mdNameField.isFirstResponder {
            let itemselected = mds[row]
            mdNameField.text = itemselected
        }
    }
}