The Weather Forecasters that work for the Met Office are paid by the Met Office, not the BBC, though the BBC will pay the Met Office for the weather service, just as they pay Camelot for the rights to broadcast the Lottery draw.
I met Micheal Fish a few years ago (I had to make replica of the weather map and magnetic clouds and sunshines for a short film) and he said when he joined the Met Office he was told to read the weather on TV, he didn't have a choice, it was part of his job. He light-heartedly said that if he had refused to read the weather on TV, he would more than likely have found himself working in a weather station in a remote part of the country...