In the UK, epidemiological modelling is done by various government bodies and universities, and it is this that is being reported in OP:
tl;dr is that we use a single datasource (not managed by us but by many dedicated others) and use case counts to estimate number of currently infectious and from that we estimate R. The running time of our R estimation calculations for all regions is around 1.5 hours on a $10/month Digital Ocean droplet.
Sometimes strange looking results on our site picked up by guests can be fed back to the maintainers of the datasource resulting in a correction of the data.
On our website, R value estimation is done by my father, a retired computer science professor with a long history of mathematical modelling experience. This is hacker news right? :)