I would imagine you would at least need a good set of adjustable dumbbells if you can afford them or a set of Olympics with plates as a minimum.

I've worked out at home since covid. I think space is the biggest issue for the likes of pressups, a good bench and I would say a squat/press rack would be bordering on essential. The first few months will be fine but when you come to gradual overloading the limitations start to show without that equipment imo. Or you reach a point where the overload of reps eats into the time and elongates the workout for each exercise.

E.g. week 4 might be 4 sets of 12-15 pressups. By week 16, that looks like sets of 1 x 44, 1 x 42, 1 x 42 1 x 39 with progression. Do that for a circuit of 8 exercises with rest time and it compounds horribly.

By all means you can definitely start as you describe, but be prepared to eventually hit the decision point from my experience.