Fix data race condition in NetworkHandler (#53788)

Summary:
Pull Request resolved: https://github.com/facebook/react-native/pull/53788

Fix thread safety issue due to member variable  in `NetworkHandler` singleton without mutex.

Also intend to un-singleton this class in the imminent future.

Changelog: [Internal]

Reviewed By: hoxyq

Differential Revision: D82460574

fbshipit-source-id: c0c614f8f1bb5ffba22872ae5717fbd2ca01f2e9