AppDelegate.h

#import <UIKit/UIKit.h>
#import <UMReactNativeAdapter/UMModuleRegistryAdapter.h>
#import <React/RCTBridgeDelegate.h>

@interface AppDelegate : UIResponder <UIApplicationDelegate, RCTBridgeDelegate>

@property (nonatomic, strong) UMModuleRegistryAdapter *moduleRegistryAdapter;
@property (nonatomic, strong) UIWindow *window;

@end

AppDelegate.m

#import "AppDelegate.h"

#import <React/RCTBundleURLProvider.h>
#import <React/RCTRootView.h>

#import <UMCore/UMModuleRegistry.h>
#import <UMReactNativeAdapter/UMNativeModulesProxy.h>
#import <UMReactNativeAdapter/UMModuleRegistryAdapter.h>

@implementation AppDelegate

- (BOOL)application:(UIApplication *)application didFinishLaunchingWithOptions:(NSDictionary *)launchOptions
{
  self.moduleRegistryAdapter = [[UMModuleRegistryAdapter alloc] initWithModuleRegistryProvider:[[UMModuleRegistryProvider alloc] init]];
  RCTBridge *bridge = [[RCTBridge alloc] initWithDelegate:self launchOptions:launchOptions];
  RCTRootView *rootView = [[RCTRootView alloc] initWithBridge:bridge moduleName:@"mynewbear" initialProperties:nil];
  rootView.backgroundColor = [[UIColor alloc] initWithRed:1.0f green:1.0f blue:1.0f alpha:1];

  self.window = [[UIWindow alloc] initWithFrame:[UIScreen mainScreen].bounds];
  UIViewController *rootViewController = [UIViewController new];
  rootViewController.view = rootView;
  self.window.rootViewController = rootViewController;
  [self.window makeKeyAndVisible];
  return YES;
}

- (NSArray<id<RCTBridgeModule>> *)extraModulesForBridge:(RCTBridge *)bridge
{
  NSArray<id<RCTBridgeModule>> *extraModules = [_moduleRegistryAdapter extraModulesForBridge:bridge andExperience:nil];
  // You can inject any extra modules that you would like here, more information at:
  // https://facebook.github.io/react-native/docs/native-modules-ios.html#dependency-injection
  return extraModules;
}

- (NSURL *)sourceURLForBridge:(RCTBridge *)bridge {
#ifdef DEBUG
  return [[RCTBundleURLProvider sharedSettings] jsBundleURLForBundleRoot:@"index" fallbackResource:nil];
#else
  return [[NSBundle mainBundle] URLForResource:@"main" withExtension:@"jsbundle"];
#endif
}
@end

For the:

NSArray<id<RCTBridgeModule>> *extraModules = [_moduleRegistryAdapter extraModulesForBridge:bridge andExperience:nil];

Isn't it better to provide some experience name instead of nil to stay on the safe side?
For example the permissions code may crash on save because of nil key. (Maybe it's fixed, but it was before for sure)

Do not forget to rename 'mynewbear' to your application name. For example: "YourApplicationName". I missed the mynewbear stuff, funny, but not helpful.

RCTRootView *rootView = [[RCTRootView alloc] initWithBridge:bridge moduleName:@"mynewbear" initialProperties:nil];

Becomes

RCTRootView *rootView = [[RCTRootView alloc] initWithBridge:bridge moduleName:@"YourApplicationName" initialProperties:nil];

Remove andExperience:nil to make it work in unimodules 0.4.1