As others have mentioned, 're-speaking' is used increasingly more for subtitling of live programmes; it's far cheaper to train a re-speaker than a stenographer.
The subtitler listens to the programme, and then repeats the dialogue, including punctuation.
Often a researcher will have reviewed the script or running order beforehand, and prepared a word list of complicated words, which the subtitler will use to train the speech recognition software (often just IBM ViaVoice).
Red Bee Media have the contract for BBC and Channel 4, amongst others. They take care of the BBC's regional news too, often remotely. I'm not sure if they still do it, but for a while, overnight subs for News 24 were originated in Sydney - Red Bee own the Australian Caption Centre.
And, since you ask, they're currently hiring: http://www.redbeemedia.com/html/current-vacancies.html
(vacancies pop up relatively frequently on that page for posts in London and Glasgow)