I have the same problem.
1x NUKI 3.0 PRO
2x AppleTV4K 2019 (NOT used as homekit hub, before 16.2 new architecture)
1x AppleTV4K 2021 (10 meters away from door, used as Homekit hub, before 16.2 new architecture)
1x Homepod mini (2 meters away from door, used only as Homekit hub for bluetooth relay to NUKI 3.0)
1x NUKI 2.0 + WIFI Bridge
1x AppleTV4K 2019 (5 meters away from door, NOT used as homekit hub, before 16.2 new architecture)
1x AppleTV4K 2021 (2 meters away from door, used as Homekit hub, before 16.2 new architecture)
In both houses AppleTVs have wired ethernet with good swicthes. WIFI (for homepod mini) is fast and strong in both houses too. Connection to internet is 1000mbit fiber. So the networks is really not the issue.
Both houses’ Homekit is controlled with iPhone 12 pro and 14pro, Watch 6 and 7, iPad Pro (M1) and Macbook pro M1pro (all Homekit New Architecture, all Apple silicon). I use my watch most with Siri (Hey Siri, open home front door).
House 2 does not really have issues with 16.2 Homekit New Architecture. That must be beacuse there are only two Homekit hubs, and both very near the NUKI 2.0 lock.
House 1 DOES have problems since 16.2 Homekit New Architecture. It’s setup worked fine with 16.1.x. The Homepod mini gave a quick Homekit (and Siri) connection to the NUKI.
Since 16.2 Homekit new architecture, I see I can no longer control in settings which AppleTVs are Homekit hubs. I suppose that means they are now all Homekit hubs, and there are now 4 Homekit hub devices (in stead of 2) searching for the NUKI via Bluetooth (because Apple only allows Nuki to communicate Homekit over Bluetooth, not via WIFI of the built in WIFI or WIFI bridge). 2 of my AppleTVs are too far from the NUKI for Bluetooth communication and one is only just is in reach of the NUKI for Bluetooth communication.
The result is terrible communication and operation of the NUKI with Homekit in House 1, since 16.2 Homekit new architecture.
The result is timeouts (Nuki lock is taking a long time to respond) and slow status updates on Homekit screens. Apple broke this, it was working reasonably (not perfect).
If only Apple gave me choice which hub to let communicate to the NUKI, or even better: Made the Homekit hubs self learning and let them remember which hub has the best Bluetooth connection to the NUKI (and other Bluetooth devices), setting that hub as default Homekit communication device for NUKI. If they do actually try to do this, that function is obviously broken.
I have submitted several bug reports to Product Feedback - Apple (selected homekit). Please, all do the same!
I also submitted the very annoying Home Switching bug (since 16.0 Homekit no longer switches between my two homes. It always defaults to House 1, even when I’m in House 2, which is far away from the other).