Evidence today shows that public transport across all modes in London is dramatically down on the levels seen just 4 weeks ago. This is good, and must continue. Please only travel if your journey is absolutely necessary, and when in the centre of London please try and keep your distance as much as possible.
The Oyster and contactless systems continue to operate normally. There shouldn’t be anything different, though I am aware of one small consideration where service frequencies have been cut. This concerns out-of-station-interchanges where, for example, tube and National Rail journeys are joined together and charged one fare. At most London terminals you have 40 minutes between touch out from the Underground and touch in at the NR gateline to allow you to wait on the concourse until your train is advertised. If your wait is longer then the two parts will be charged separately and will probably cost a little more. To avoid this you might want to consider travelling to another station and waiting for your train there, for example Clapham Junction from Victoria or Waterloo, or London Bridge from Cannon Street or Charing Cross. As long as your wait is while touched in you won’t end up paying two fares.