Have each phone generate a public/private key pair, and upload the public key to an anonymous repository.

To escrow a key the client app uses a secret sharing system to generate 10000 key shares, with any 500 key shares being sufficient to reconstruct the escrow key. It encrypts each share with a randomly chosen public key from the anonymous repository. It sends these encrypted shares to the government.

If the government wants to recover the escrow key, it has to make a public request for help in which it publishes the 10000 encrypted key shares. Have an app on the phone that can receive notification of such requests, and if it sees that your public key was used for one of the shares it asks you if you approve the government's request to recover the escrow key, and presents the government's argument in favor of this.

If you approve, the app on your phone decrypts the share, encrypts it with a government key, and uploads it anonymously to the government.

In effect, this system provides a jury of 10000 randomly chosen anonymous phone users to review the access request, and 500 of them have to approve in order for access to be granted.