I've forgotten most of the history I once knew about all of this, you all seem to have a pretty good grasp on the timelines.
I'll just point out that it had to be obviously "possible", the only question being a matter of engineering details related to the practicality of construction, efficiency, exact modes of operation, etc.
Ever since before the Curies' works energetic radiation from various elements was known based upon its effects on film, generating Roentgen rays (x-rays) which were able to pass through solid matter, et. al. Quantum theory even in the early 1900s certainly would have known that the X-rays could be a form of EM radiation and that it'd be more energetic than Ultraviolet, et. al. So nuclear decomposition that produced such energetic particle or wave outputs had to be a powerful process.
I'm at least somewhat suspicious that there would have also been early observations of some kinds of fission effects having to do with natural emissions which changed in character or became more intense as greater quantities of the substance were assembled or one radioactive substance was permitted to irradiate other substances et. al.
Also I don't think it could have been conceived that one would be able to know about elementary quantum physics, know that E=M*c^2 and not be pretty confident that it must be some kind of mass / energy conversion process that was responsible for stellar energy production since clearly the science of the 1800's would have shown that it couldn't sensibly be any merely chemical process.
And even by simple chemical analogy it would be obvious that given sufficient activation energy (heat / radiation) for a process you could decompose any given substance into its constituents, and if there was substantial energy tied up in that substance's formation it would have to be released in the process.
So given good knowledge of natural unstable radioisotopes, having some conception of the nature of their unstable decay into lighter elements, some conception of the high energy involved (the HEAT was noticable even to the Curies, et. al., as well as the x-rays) , and seeing that stars generated extreme energies without quickly burning through much of their mass, it'd be fairly obvious as a physical possibility and fairly also obvious how to proceed to experimentally and theoretically investigate the processes using natural radioisotopes, et. al.