Reconnecting iPad after iTunes have forgotten it without wiping contents
It seems I assumed wrong. Thanks to kenansulayman's answer, I dared to continue after the "Set Up Your iPad" screen. When I did, iTunes did not wipe the iPad, it left the old configuration unaltered and started syncing normally.

Also, "moving" the iPhone was fairly straightforward. I just unchecked the "Sync with this iPhone over Wi-Fi" on the old computer, synced it with the new and checked "Sync with this iPhone over Wi-Fi" there.