ClubSpark (which operates the court bookings system on behalf of the LTA) has come up with a temporary workaround which will offer you a one time link. See https://clubspark.lta.org.uk/ReigateLTC/Account/RequestPasswordlessSignIn?returnUrl=https://clubspark.lta.org.uk/ReigateLTC/Booking/BookByDate&source=ReigateLTC

Alternatively, the app may still be working for some users.

Nobody has access to LTA logins at the moment.