Thank you for your detailed answer. Indeed there was a typing mistake in my first sentence.No, it's not correct, but I presume you mean:
You can watch the series at 7.00 on BBC.
You use 'on'; and 'the' for the network: 'on the BBC' if it's on any one of the BBC stations. For a particular station, 'on BBC 2', 'on BBC Radio 3'. If a network has only one (relevant) station, you can treat the network name as the station name: 'on ITV', 'on Fox'. I can't think of when you would say 'at' or 'in' with a station.