I have some Sony headphones from a decade ago with a detachable cable. Noise cancellation works just fine when wired, and you get better battery life since the Bluetooth part isn’t active. The only time you can’t use noise cancellation is when it’s charging (Micro-USB, doesn’t do audio over USB in case you were wondering).
Re: noise canceling... recently got a pair of IEMs (Etymotic ER2XR) with good foam tips and their isolation blows away any ANC I've ever tried. The only thing is noise from touching the cable but that was solved with some ear hooks to put the cables behind my ears.