While weighted dips would be a suitable 'press' variation, I feel you'd have a hard time replacing the deadlift with an effective bodyweight alternative.
It depends on what you're looking to achieve though. Pull-ups and dips (especially with added weight) would form an effective routine for the upper body. All that's missing is lower body/posterior chain work.
I meant as a substitute. Obviously no better strength lift than the deadlift, and probably will never do this program with other lifts, but if it suited your needs better would it work with various callisthenics movements like dips, pull ups, pistol squat, hip bridges, handstand push ups....?
As a secondary question which bodyweight lifts would work best?