I asked the same question of the RSPB and that was their reply.
There is no problem with putting a camera in a nestbox or on a feeder.
Because the cameras will be stationary they will not disturb the birds.
I would recommend a B/W camera for the nestboxes initially because they will operate during the day without any extra illumination. If you opt for a colour one then light is limited inside the box and you would need some white light. If you use a small bulb or white LED's then ensure that they are turned off at night time so that the birds can recognise night and day. You could add some invisible infra red LED's if you wanted to see what was happening at night.