I have a bird box camera with a cable. The problem you will have is power. A cable supplied camera is tiny about half the size of a matchbox.
To get enough power the battery will make the camera quite large. I’m sure they exist but I’d imagine they are expensive and don’t have a great battery life, days rather than weeks, this would mean disturbing the nest to change the battery which would scare the birds off.
My camera came with a 20m power lead which is very thin 5v cable, maybe 2/3mm in diameter. The camera is wireless with a little receiver at the tv end.
Enyaq EV