Oyster will be accepted at Epsom from Monday 25th February. Details are a bit sketchy at present, especially as the fares are not yet in the publicly available fare-finder. However, we do have a quote from TfL:
Shashi Verma, Chief Technology Officer at Transport for London said:
We are delighted that, working with the DfT, GTR and South Western Railway, we have now been able to expand Oyster and contactless to cover services to Epsom station. Pay as you go with contactless and Oyster has helped revolutionise travel on Tube, rail and bus services across London, making them more convenient for all. We are now working with the DfT and train operating companies on expanding pay as you go to more stations across the GTR network, including Hertford North, Welwyn Garden City and Luton, throughout 2019.