This has been solved
See full discussion over at reddit here.

After upgrading to iOS 16 (on phone, homepods and atvs) and performing the “underlying architecture upgrade” my flic buttons stopped working.

In the home app the action flashed when the flic button was pressed so it seemed like physical comms were ok.
Only thing is the target of the action (turn a light on) did not happen.

So I figured best to start over. I removed all the buttons from homekit via the flic app.
I also removed the flic hub via the home app under hubs and bridges.

Finally I reset the flic hub with its pin hole and re-paired it, added a button and enabled homekit in the flic app.

The bridge is listed again in the home app now and it reports (1) accessory, which is good! screenshot

The problem is when I open the accessories list for the bridge, no buttons show despite the (1) on the list view. screenshot

The button does not show in any rooms either.
It is driving me a little crazy

Also it means my flic buttons are unusable in homekit.
Anyone else seeing this issue or have any thoughts on what to try next?