If you eat right, use correct form with bodyweight exercises (so for pull ups, chest to bar with a slight pause at the bottom in the hang and at the top,, for push ups, fully protract the shoulders at the top, and go down until your chest touches the ground), and are prepared to do harder variations of the exercises when they become too easy (you can do 15+ reps in a set), you can go very far.

As for the routine I would take out sit ups and replace it with hanging leg raises or dead bugs if you really want to do an ab exercise. Just remember that spot reduction of fat is a myth and not real. Sit ups involve spinal flexion and can put a lot of pressure on the lumbar spine.

Push ups and pull ups are good, but I'd recommend adding in a horizontal pull like a bodyweight/inverted row, a overhead or vertical push like a pike push up and to train legs. A squat variation that can challenge you with only your bodyweight, like a split squat, and something for the hamstrings and posterior leg muscles, like a single leg Romanian deadlift or a Nordic curl. You don't have to do all these exercises in one day, I would recommend splitting it up over two or more days if you need.

You don't need to use weights so long as you are prepared to train at the correct intensity with bodyweight. The hardest part to train with bodyweight only is definitely legs, but unless you can do something like 3 sets of 20 pistol squats that break parallel there is still plenty you can do with bodyweight only.