No cool air from either front or rear vents

If I am correct, your mode is on a button, and knobs control the driver and passenger temp. It should not take that long for the system to update. It should take minutes. There is an actuator recalibration procedure.

2002 Cadillac Escalade | Avalanche, Escalade, Suburban, Tahoe, Yukon (VIN C/K) Service Manual | HVAC | HVAC Systems - Automatic | Diagnostic Information and Procedures | Document ID: 774485

Actuator Recalibration
The HVAC control module will relearn the actuator limits whenever power is disconnected and re-connected to the module.

Use the following steps to perform the calibration update:

Turn OFF the ignition.
Remove the battery positive voltage circuit fuse of the HVAC Control Module.
Important: The scan tool must be disconnected from the vehicle to properly perform the calibration procedure. If a scan tool is connected before this procedure is completed, then new calibration values will not be stored.

Important: Failure to wait 60 seconds will not allow the HVAC control module sufficient time to clear the old calibration values.

Wait 60 seconds.
Install the fuse.
Turn ON the ignition.
Wait 60 seconds before installing a scan tool.

There is also a couple bulletins that may apply.

#04-01-39-007: Loss of HVAC System Mode and/or Temperature Control (Replace HVAC Control Module) - (Jun 17, 2004)

Subject: Loss Of HVAC System Mode and/or Temperature Control (Replace HVAC Control Module)

Models: 2002 Cadillac Escalade, Escalade EXT

1999-2002 Chevrolet Silverado, Suburban, Tahoe

2002 Chevrolet Avalanche

1999-2002 GMC Denali, Sierra, Yukon, Yukon XL

2001 GMC Sierra C3

2002 GMC Sierra Denali

with Automatic HVAC System (RPO C68)

Condition
Some customers may comment that they cannot control the HVAC system modes and/or the temperature. These concerns may be intermittent.

Cause
An HVAC control module logic lock-up may be the cause of these conditions.

Correction
Important: Prior to replacing the HVAC control head, review Corporate Bulletin Number 01-01-39-007 (SI Document ID #845221).

Replace the HVAC control module. Refer to the HVAC Control Module Replacement procedure in the HVAC System -- Automatic sub-section of the Service Manual.

Part Information
Part Number
Description

15126606

Condition
Some customers may comment that they cannot control the HVAC system modes and/or the temperature. These concerns may be intermittent.

Cause
There may be poor terminal contact and/or bent terminals in connectors C206 (RPO C68 automatic HVAC controls) or C296 (RPO C60 manual HVAC controls). Excessive tension on the wires due to the wiring harness being misrouted in the instrument panel carrier may cause these conditions.

Correction
Remove the Air Bag fuse.
Disconnect the steering wheel and instrument panel (IP) air bag connectors. Refer to the SIR subsection of the Restraints section of the Service Manual for connector locations.
Remove the IP upper trim pad.
Locate connector C206 (6 cavities) or C296 (8 cavities). The connector is attached to a metal brace on the far right side of the IP carrier above the IP air distributor duct.
Inspect the routing of the wiring harness to the connector. The proper routing of the wiring harness is above the air distributor duct with no tension on the terminals in the connector. The routing is improper if the wiring harness is below the air distributor duct with excessive tension on the terminals in the connector. The excessive tension may cause poor terminal contact, resulting in open circuits, in the connector. The wires may also pull the terminals out of the connector or pull out of the terminal itself.
If the harness is routed properly, contact Technical Assistance for further information.
If the harness is misrouted, remove the connector from the brace.
Disconnect the connector.
Perform a pin drag test on the female terminals.
Inspect the male terminals for straightness.
Inspect the wire to terminal connection to insure that no wires have pulled out of the terminals.
Reroute the wiring harness above the air distributor duct.
Firmly reconnect and reseat the connector.
Reattach the connector to the metal brace.
Reinstall the IP upper trim pad.
Turn Off the ignition and reconnect the steering wheel and IP air bags.
Reinstall the Air Bag fuse.
Turn ON the ignition and verify that the air bag light flashes seven times and goes out.
Clear any diagnostic trouble codes that may have been set.
Verify proper operation of the HVAC system mode and temperature controls.

Do you have the air conditioner set to auto when the temperature change happens or on manual mode?
Does the above concern happen more frequently when the vehicle is idling and not so much when driving? The manufacture has release a technical service bulletin on the cooling fans not cooling sufficiently when idling for extended period of time such as sitting in traffic.

This guide can help

https://www.2carpros.com/articles/car-air-conditioner-not-working-or-is-weak

Please run down this guide and report back.

Intermittent issues can be one of the worst diagnostic jobs given in my profession! Obviously the key here is get the a/c system to act up all the time so a proper diagnostic can happen. Unfortunately there are so many variables that can cause the above concern to happen. Is it a mechanical or electrical issue? A Ambient (out side the vehicle) temperature sensor signal can drop out (.-40Deg.) Causing the climate control module to shut down the climate control. A restriction in the refrigerant pipes could be causing a intermittent freeze up (internally). Broken temp blend door sticking open enough to allow hot air through. Temp blend door actuator internal gears worn causing door to stay open. Faulty wiring to a pressure sensor signal or any input or output wiring to control module
I would first try to pull any diagnostic fault codes from the climate control module to see if there are any code or history codes (the code will start with a B [or bxxxx] followed by four numbers). You'll need to get a hold of someone with a factory compatible scan tool to do this and access the module. If there are any codes this would certainly narrow down where to start testing.
As I mentioned before there is a TSB (reference number 01-01-39-004) which describes a loss in a/c cooling and the correction is to add a auxiliary cooling fan to increase air flow through condenser to assist in removing more heat.
Hope this helps you, TY

Sounds like a possible temp door actuator fault, or HVAC control module. Does it change temperature at all, or does it seem stuck in one position?

I have access to about 26 images for the automatic air condition system. Most of them are wiring diagrams. Is there something specific your looking for?

If you have a voltmeter, and know how to use it, you can reference the following fault chart.
Temperature Actuator Inoperative (FRONT)
Step

Action
Value(s)
Yes
No

1
Has the HVAC Air Delivery System Check been performed?
--
Go to Step 2
Go to HVAC Air Delivery System Check

2
Place the ignition switch in the RUN position.
Disconnect the connector at the front temperature door motor.
Connect DMM (volts) between the front temperature door motor connector, cavity 10, and ground.
Is battery voltage present?
--
Go to Step 4
Go to Step 3

3
Locate and repair the open in CKT 141 (BRN) between the IP fuse block and the front mode door motor. Refer to Wiring Repairs in Wiring Systems.

Is the repair complete?
--
Go to HVAC Air Delivery System Check
--

4
Connect a DMM (continuity) between the front temperature door motor connector, cavity 7, and ground.

Is continuity present?
--
Go to Step 6
Go to Step 5

5
Locate and repair the open in CKT 150 (BLK) between the front temperature door motor and splice S244. Refer to Wiring Repairs in Wiring Systems.

Is the repair complete?
--
Go to HVAC Air Delivery System Check
--

6
Place the ignition switch in the RUN position.
Place the blower switch on the HVAC control module in the LO position.
Make sure that the front blower motor is operating.
Connect a DMM (volts) from the front temperature door motor connector, cavity 8, to ground.
Adjust the position of the temperature control dial while observing the DMM response.
Is the DMM response a smooth, variable voltage?
--
Go to Step 9
Go to Step 7

7
Disconnect connector C3 at the HVAC control module.
Connect a DMM (volts) between pin 8 of the HVAC control module, and ground.
Adjust the position of the temperature control dial while observing the DMM response.
Is the DMM response a smooth, variable voltage?
--
Go to Step 8
Go to Step 10

8
Locate and repair the open or cause of resistance in CKT 733 (LT BLU) between the HVAC control module and the front temperature door motor. Refer to Wiring Repairs in Wiring Systems.

Is the repair complete?
--
Go to HVAC Air Delivery System Check
--

9
Replace the front temperature door motor. Refer to Temperature Actuator Replacement .

Is the repair complete?
--
Go to HVAC Air Delivery System Check
--

10
Replace the HVAC control module. Refer to Control Assembly Replacement .

Is the repair complete?
--
Go to HVAC Air Delivery System Check

I am not sure if the temperature actuator can be accessed in situ, for replacement it shows partial removal of the instrument panel. Here is a view.

(1) Blower Resistor
(2) Blower Motor
(3) Blower Motor Connectors
(4) C230
(5) C203
(6) C204
(7) Temperature Door Motor
(8) HVAC Plenium
(9) Mode Door Motor
(10) HVAC Wiring Harness
(11) Blower Motor Relay, High

The HVAC control module is the same as the control panel.