SignButtons에는 loading Props를 설정해주세요. 다음으로 SignButtons에서 loading 값이 true일 때 스피너를 보여주는 ActivityIndicator 컴포넌트가 나타나도록 구현하세요.

components/SignButtons.js

import React from 'react';
import {ActivityIndicator, StyleSheet, View} from 'react-native';
import CustomButton from '../components/CustomButton';
import {useNavigation} from '@react-navigation/native';

function SignButtons({isSignUp, onSubmit, loading}) {
    
  (...)

  if (loading) {
    return (
      <View style={styles.spinnerWrapper}>
        <ActivityIndicator size={32} color="#6200ee" />
      </View>
   );
  }

  return (...);
}

const styles = StyleSheet.create({
  spinnerWrapper: {
    marginTop: 64,
    height: 104,
    justifyContent: 'center',
    alignItems: 'center',
  },
  buttons: {
    marginTop: 64,
  },
});

export default SignButtons;