The human eye has a resolution of roughly 2 arc minutes, or about 0.5 milliradians (15 pixels across a full moon). The distance to Betelgeuse is a bit uncertain, but lets say 600 light years, which is near enough. So to show a disk simply because of the size of the source, it would need to be about 0.3 light years across.
The source can't grow faster than the speed of light, and it reasonably might grow that fast as the radiation from the supernova heats the dust and gas already shed by Betelgeuse. If so it would take about 0.15 years or about 8 weeks to grow large enough to (just) show a visible disk. The actual material ejected in the supernova explosion would move at a fraction of that velocity.
One thing that might happen is that scattering in the Earth's atmosphere might make a very intense point source look larger. Light would arrive at your eye from a number of directions close to the point source, having scattered on its way through the atmosphere. Effects in the eye might also make a very bright point source look larger.