Restore "Ready for new phone" back up to new phone

Work has issued me with an iPhone 13 to replace my existing iPhone 7.

The process to create an 'apps & data' back up from my old phone was excellent, and getting the two phones to talk to each other was also smooth & quick. My iCloud storage is full but Apple allows extra capacity for 21 days to accommodate migration from one phone to another.

At no point however did I get a "Transfer Your Apps & Data" screen. I expected to see this after reaching the "Finish on New iPhone" screen, and setting up wi-fi.

Is there any way I can manually tell the new iPhone to look for the special iCloud back up?

Our IT guy thinks it is impossible, but that seems bizarre to me, that Apple would have this special back up without a way to manually pull it into the process. There's certainly a lot of process and security around the whole set up and identifying myself. I would have thought that would be enough to ensure such a transfer wasn't disqualified for security reasons.

The IT guy reset my iPhone 13 and we started the new phone set up from scratch again, confirming the "Transfer Your Apps & Data" screen was not showing at any point.