Transport for London have a fairly comprehensive API that has all kinds of live journey and network information: https://api.tfl.gov.uk/ - the article mentions that this project is using the API
Edit: Looks like TMB also have an API. I'm not sure how it compares to TfL's: https://developer.tmb.cat/
Thanks! Yes TMB has an API, I looked at it before but it only provided scheduled service times, not live data like you can see on the boards in the station. But I'll have another look!
Edit: Looks like TMB also have an API. I'm not sure how it compares to TfL's: https://developer.tmb.cat/