Получать ответ от поп-навигатора во Flutter

Добавьте ожидание, чтобы получить результат от Future и вернуть значение;

   Future<String> getFormData(BuildContext context) async {
    final result = await Navigator.push(
        context, MaterialPageRoute(builder: (context) => MyCustomForm()));
    return Future.value(result);
  }

Измените код FloatingActionButton onPressed для получения Future String

onPressed: () async {
            final value = await getFormData(context);
            setState(() {
              data = value;
            });
          },

А вот код, если вы хотите передать объект данных, а не строку:

main.dart:

import 'package:flutter/material.dart';

import 'answer.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    final appTitle = 'Form Validation Demo';

    return MaterialApp(
      title: appTitle,
      home: Scaffold(
        appBar: AppBar(
          title: Text(appTitle),
        ),
        body: ShowData(),
      ),
    );
  }
}

// Create a Form widget.
class MyCustomForm extends StatefulWidget {
  @override
  MyCustomFormState createState() {
    return MyCustomFormState();
  }
}

// Create a corresponding State class.
// This class holds data related to the form.
class MyCustomFormState extends State<MyCustomForm> {
  // Create a global key that uniquely identifies the Form widget
  // and allows validation of the form.
  //
  // Note: This is a GlobalKey<FormState>,
  // not a GlobalKey<MyCustomFormState>.
  final _formKey = GlobalKey<FormState>();
  final myController = TextEditingController();
  Data stateData = Data();
  @override
  void dispose() {
    // Clean up the controller when the widget is disposed.
    myController.dispose();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    // Build a Form widget using the _formKey created above.
    return Scaffold(
      body: Form(
        key: _formKey,
        child: Column(
          crossAxisAlignment: CrossAxisAlignment.start,
          children: <Widget>[
            TextFormField(
              controller: myController,
              validator: (value) {
                if (value.isEmpty) {
                  return 'Please enter some text';
                }
                return null;
              },
              onSaved: (value){
                stateData.load = value;
              },
            ),
            Padding(
              padding: const EdgeInsets.symmetric(vertical: 16.0),
              child: RaisedButton(
                onPressed: () {
                  // Validate returns true if the form is valid, or false
                  // otherwise.
                  if (_formKey.currentState.validate()) {
                    _formKey.currentState.save();
                    // If the form is valid, display a Snackbar.
                    Navigator.pop(context,stateData);
//                  Scaffold.of(context)
//                      .showSnackBar(SnackBar(content: Text(myController.text)));
//                    myController.text = 'look at me';
                  }
                },
                child: Text('Submit'),
              ),
            ),
          ],
        ),
      ),
    );
  }
}

class Data {
  String load;
}

answer.dart:

import 'package:flutter/material.dart';

import 'main.dart';

class ShowData extends StatefulWidget {

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

class _ShowDataState extends State<ShowData> {
  String data = 'start';

  @override
  Widget build(BuildContext context) {
    return Scaffold(
        floatingActionButton: FloatingActionButton(onPressed:   () async {
          final holder = await  getFormData(context);
          setState(() {
            data = holder.load;
          });
        },
        elevation: 4,

        ),
        body:Text(data,style: TextStyle(fontSize: 80),));
  }

  Future<Data> getFormData(BuildContext context) async {
    final answer =  await Navigator.push(context,MaterialPageRoute(builder: (context)=>MyCustomForm()));
        return (Future.value(answer));
  }
}