Seems like transactions could not be initiated from my wallet (iOS). Those just didn't get shared. Handing over a PSBT also didn't work. It did work finally when my co-signer initiated them from the desktop app. The transaction popped up in our chat and I could sign it with a few taps. That signed status was not properly reflected in the other wallet though.

Nunchuk needs more testing and better error messages when things don't work out. Very nerve-wrecking to be in a situation where you have worry if you can ever spend your funds at all, and this is the key feature why people use Nunchuk.