Replaced factory radio with another factory radio, the signal lights flashed and the engine will not crank?

Typical GM issue. The radio in them is not just the radio, it also is part of the security system. When you swap them, you need to use an OE level scan tool to go in and program the VIN into the replacement unit and match the option coding in the body module to enable the replacement radio. Then once it is set up you need to relearn the keys into the system so that it will start and run. GM integrated it to help prevent theft and it causes problems for the owners. Depending on what is wrong with your old unit you may want to look into having it repaired.

Signal lights flashing and engine won't turn over.

Hello, in this case, the replacement radio needs to have the VIN number of the current vehicle programmed into it, with also most likely a software update, there are always updates. If all the modules in the vehicle do not have the same software updates, you can end up getting a "Checksum error" trouble code setting. Programming any module requires a capable scan tool. The Radios are usually tied into the security system because they are often stolen. So, it could cause a no start condition. You may want to try putting in the old radio for now and see if it will start.