Metallurgical Homogeneity

The variations in composition will come from carbon and alloy segregation inherited from the solidification structure,and modified by the soaking and hot working practice used to make the shaft.
Variations in hardness will come from how heat treatment and resultant microstructure are affected by the composition variations. You are at a large enough diameter to make this difficult to predict, since you are beyond Jominy bar cooling rates. Actually, you are asking for a whole course in physical metallurgy of steel!
Some insight into composition variations can be gotten by looking at standards such as SAE Standard J409 "Product Analysis - Permissible Variations from Specified Chemical Analysis of a Heat or Cast of Steel". The variations depend both on the element and its specified range though, even at a given cross sectional area. You can find J409 in a copy of the SAE Handbook. Look at your friendly local engineering library.
Pittsburgh Pete

Would ASTM A914 be of use? This gives the expected range of hardness (presumably mostly variation between casts), at the surface,the centre and an intermediate position for some alloys, I can not remember if it goes to large enough diameters.