Single Instance of Widget class is used even when different variables are coded to be used

There are several issues with your code:

1- The real problem is that you never rebuild the BaseWidget. You construct 3 new BaseWidgets, but you only ever call the build of the _home widget, because it's the first one returned by choosePager(). Since you don't create _home, _business, _school in the HomeWidget build, no other BaseWidget can ever get built.

2- When you don't need to store any state/variables for a widget, use a Stateless widget.

3- Don't do anything in the constructor of your State. Use initState https://docs.flutter.io/flutter/widgets/State/initState.html for that instead.

4- Create widgets using const constructors when possible.

5- Widget constructor take named parameters. One of those should be the key. Use super to call the base constructor.

With that in mind, this is what the code should look like:

class BaseWidget extends StatelessWidget {
  final String title;

  const BaseWidget({Key key, this.title}) : super(key: key);

  @override
  Widget build(BuildContext context) {
    return Center(
      child: Text(title),
    );
  }
}

class HomeWidget extends StatefulWidget {
  _HomeWidgetState createState() => _HomeWidgetState();
}

class _HomeWidgetState extends State<HomeWidget> {
  int pageIndex = 0;

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      home: Scaffold(
        appBar: AppBar(
          title: Text('Bottom Navigation Bar'),
        ),
        body: choosePager(),
        bottomNavigationBar: BottomNavigationBar(
          currentIndex: pageIndex,
          items: <BottomNavigationBarItem>[
            BottomNavigationBarItem(
              icon: Icon(Icons.home),
              title: Text('Home'),
            ),
            BottomNavigationBarItem(
              icon: Icon(Icons.business),
              title: Text('Business'),
            ),
            BottomNavigationBarItem(
              icon: Icon(Icons.school),
              title: Text('School'),
            ),
          ],
          onTap: onTap,
        ),
      ),
    );
  }

  void onTap(int index) {
    setState(() {
      pageIndex = index;
    });
  }

  Widget choosePager() {
    Widget result;

    switch (pageIndex) {
      case 0:
        result = BaseWidget(title: 'Home');
        break;
      case 1:
        result = BaseWidget(title: 'Business');
        break;
      case 2:
        result = BaseWidget(title: 'School');
        break;
      default:
        result = Text('Unknown');
        break;
    }
    return result;
  }
}

Edit: For your example, you may want to fetch some data from the network and only use the widget to display it. In that case, create a new class (not a Widget) to fetch & hold on to the data, and use the Widget only for displaying the data.

Some sample code:

/// Silly class to fetch data
class DataClass {
  static int _nextDatum = 0;
  int _data;

  DataClass();

  Future<int> fetchData() async {
    await Future.delayed(Duration(
      milliseconds: 2000,
    ));
    _data = _nextDatum++;
    return _data;
  }

  int getData() {
    return _data;
  }
}

class BaseClass extends StatefulWidget {
  final String title;
  final DataClass data;

  const BaseClass({Key key, this.title, this.data}) : super(key: key);

  @override
  _BaseClassState createState() => _BaseClassState();
}

class _BaseClassState extends State<BaseClass> {
  String title;
  DataClass data;

  @override
  Widget build(BuildContext context) {
    String dataStr = data == null ? ' - ' : '${data.getData()}';

    return Center(
      child: Text(
        '$title: $dataStr',
      ),
    );
  }

  @override
  void initState() {
    super.initState();
    // initState gets called only ONCE
    title = widget.title;
    data = widget.data;
  }

  @override
  void didUpdateWidget(BaseClass oldWidget) {
    super.didUpdateWidget(oldWidget);
    if (widget.data != oldWidget.data) {
      data = widget.data;
    }
    if (widget.title != oldWidget.title) {
      title = widget.title;
    }
  }
}

class HomeWidget extends StatefulWidget {
  _HomeWidgetState createState() => _HomeWidgetState();
}

class _HomeWidgetState extends State<HomeWidget> {
  int pageIndex = 0;
  Map<String, DataClass> _dataMap = <String, DataClass>{};

  @override
  void initState() {
    super.initState();
    _init().then((result) {
      // Since we need to rebuild the widget with the resulting data,
      // make sure to use `setState`
      setState(() {
        _dataMap = result;
      });
    });
  }

  Future<Map<String, DataClass>> _init() async {
    // this fetches the data only once
    return <String, DataClass>{
      'home': DataClass()..fetchData(),
      'business': DataClass()..fetchData(),
      'school': DataClass()..fetchData(),
    };
  }

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      home: Scaffold(
        appBar: AppBar(
          title: Text('Bottom Navigation Bar'),
        ),
        body: choosePager(),
        bottomNavigationBar: BottomNavigationBar(
          currentIndex: pageIndex,
          items: <BottomNavigationBarItem>[
            BottomNavigationBarItem(
              icon: Icon(Icons.home),
              title: Text('Home'),
            ),
            BottomNavigationBarItem(
              icon: Icon(Icons.business),
              title: Text('Business'),
            ),
            BottomNavigationBarItem(
              icon: Icon(Icons.school),
              title: Text('School'),
            ),
          ],
          onTap: onTap,
        ),
      ),
    );
  }

  void onTap(int index) {
    setState(() {
      pageIndex = index;
    });
  }

  Widget choosePager() {
    Widget result;

    switch (pageIndex) {
     // it doesn't matter if you create a new BaseClass() a hundred times, flutter is optimized enough to not care. The `initState()` gets called only once.  You're fetching the data only once.
      case 0:
        result = BaseClass(
          title: 'Home',
          data: _dataMap['home'],
        );
        break;
      case 1:
        result = BaseClass(
          title: 'Business',
          data: _dataMap['business'],
        );
        break;
      case 2:
        result = BaseClass(
          title: 'School',
          data: _dataMap['school'],
        );
        break;
      default:
        result = Text('Unknown');
        break;
    }
    return result;
  }
}

Please try this code, I have edited the BaseWidget class

  class BaseWidget extends StatefulWidget {
  final String title;

  BaseWidget(this.title);

  _BaseWidgetState createState() => _BaseWidgetState();
}

class _BaseWidgetState extends State<BaseWidget> {


  @override
  Widget build(BuildContext context) {
    return Center(child: Text(widget.title));
  }
}