Transceiver has been replaced but scope says that the transceiver is not receiving the input.

Forget the other flash codes, this is the main one you need to diagnose an fix.

B1681 PATS Transceiver Module Signal is Not Received PCM GO to Pinpoint Test E .
Do you know how to do electrical testing using a Voltmeter?

E2 CHECK THE PATS TRANSCEIVER FOR VOLTAGE CIRCUIT 16 (RD/LG)
PATS Transceiver C2097
Measure the voltage between the PATS transceiver C2097 pin 2, circuit 16 (RD/LG), harness side and ground.

• Is the voltage greater than 10 volts?

Yes GO to E3 .

No REPAIR the supply circuit. CLEAR the DTCs. TEST the system for normal operation

What Does the B1681 Ford Code Mean? During each vehicle start sequence, the transceiver module reads the Passive Anti-Theft System (PATS) key identification code and sends the data to the Powertrain Control Module (PCM). B1681 Ford Code - PATS Transceiver Module Signal Is Not Received

When you disconnected the battery you also disconnected all the systems for the immobilizer and security system. It needs resetting.

You can't just change parts an expect it to work , software , modules won't let it work . You have to take it to the dealer or have a mobile lock smith fix it , reprogram it ETC...
Anti-Theft -Passive Anti-Theft System (PATS) The passive anti-theft system (PATS) contains the following components:

• theft indicator
• encoded ignition key
• PATS transceiver module
• powertrain control module (PCM)
• standard corporate protocol (SCP) communication network
• Passive Anti-Theft System (PATS) -Diagnostic Trouble Codes (DTC) Index Powertrain Control Module Diagnostic Trouble Code (DTC) Index DTC Description Source Action B1213 Anti-Theft Number of Programmed Keys is Below Minimum PCM GO to Pinpoint Test F . B2431 Transponder Program Failure PCM VERIFY if using correct PATS key, if defective USE new key. B1342 ECU is Defective (EEFROM in PCM not working) PCM INSTALL a new PCM. REFER to Section 303-14 . B2103 Antenna Not Connected PCM GO to Pinpoint Test A . B1600 PATS Ignition Key Transponder Signal is Not Received PCM GO to Pinpoint Test B . B1601 PATS Received Incorrect Key-Code From Ignition Key Transponder PCM GO to Pinpoint Test C . B1602 PATS Received Invalid Format of Key-Code From Ignition Key Transponder PCM GO to Pinpoint Test D . B1681 PATS Transceiver Module Signal is Not Received PCM GO to Pinpoint Test E . - All Other DTCs PCM REFER to Section 419-10 .

How are you coming up with a code 17 ?
PATS Diagnostic Trouble Code (DTC) Index
PATS Diagnostic Trouble Code (DTC) Index DTC Description Action B1213 Anti-Theft Number of Programmed Encoded Ignition Keys Below Minimum GO to Pinpoint Test B1213 . B2103 or B1232 Antenna Not Connected Defective Transceiver GO to Pinpoint Test B2103 . B1600 No Key Code Received - Damaged Encoded Ignition Key or Use of Non-PATS Encoded Ignition Key GO to Pinpoint Test B1600 . B1601 Unprogrammed Encoded Ignition Key (Keycode Format OK) GO to Pinpoint Test B1601 . B1602 Invalid Key Code Format From Encoded Ignition Key Transponder (Partial Key Read) GO to Pinpoint Test B1602 . B1681 PATS Transceiver Signal Is Not Received (Not Connected, Damaged, or Wiring) GO to Pinpoint Test B1681 . B2139 PCM ID Does Not Match Between PATS and PCM GO to Pinpoint Test B2139 . B2141 No Security ID Exchange Between PATS and PCM GO to Pinpoint Test B2141 . U1147 Faulty SCP Link or Incorrect PCM Calibration GO to Pinpoint Test U1147 . U1262 Missing SCP Message REFER to Section 418-00 .

Try www.reliable-store.com , it's very helpful in repairing your
vehicle

I believe the transceiver ring is more of an antenna, so it most likely require programming. Some locksmiths can do that now.

That box is for the anti-theft system , he changed the lock cylinder . You need new PATS keys . PATS - passive anti theft system .The transceiver module communicates with the encoded ignition key. This module is located behind the steering column shroud and contains an antenna connected to a small electronics module. During each vehicle start sequence, the transceiver module reads the encoded ignition key identification code and sends the data to the instrument cluster.
And the dealer is the only one that can program the keys . or a mobile lock smith .

Did you put PATS transceiver module out of old steering into your replacement column ?
The transceiver module communicates with the encoded ignition key. This module is located behind the steering column shroud and contains an antenna connected to a small electronics module. During each vehicle start sequence, the transceiver module reads the encoded ignition key identification code and sends the data to the instrument cluster.
Other wise you'll have to take to the dealer to be reset , can only be done with dealer scan tool .

To Reprogram A Single Button

1. Press & hold the desired universal transceiver button. Do not release the button until Step 4 has been completed.
2. When the dot(s) under the HomeLink icon being to flash slowly (after 20 seconds), position the hand-held transmitter 1-3" away from the universal transceiver surface.
3. Press & hold the hand held transmitter button.
4. The universal transceiver light will flash, first slowly and then rapidly. When the indicator light begins to flash rapidly, release both buttons.

The previous programming for the button has now been erased and a new device has been programmed to the button.

Repeat for any other buttons you want to do for other devices.