How lovely, what a great project.
It looks a lot thinner than 3/4" in those photos. Personally I'd laminate this, but I guess you could try steam bending. The lower of the two components that form the "S" shaped spine would be easier because it's secured at both ends. Spring back on the top most component would be trickier because it's free to waggle around at the top end in any way it wants (which is why I'd laminate). The base unit/feet and other components are all cut from the solid, you'd need perfectly dry wood for these parts or as it dries they will twist and the stand will then rock.
To answer your question, 24 hours in the form is the very minimum, a week is better.
Yeah, thinking again I'd definitely laminate. Glue lines wouldn't be too much of an issue because the edges of the curved components are square, in other words you're not overly exposing the glue lines by cutting a curved profile on the edges. Plus lamination is the best way to get a really strong, rigid, but thinner component. The UF glue you'd use for lamination dries glass hard, so imparts it's own stiffness into a structure. Take the top spine component down to a steam bent 1/2" and it would be awfully whippy, make it 3/4" and it might look clunky and top heavy.
There's a lot of work in this project, don't be driven by the fact you happen to have some green Oak on hand, there's only a few quid of timber needed for this project so material costs are totally negligible compared to your time and effort.
Just my top of head thoughts.
Good luck!