I am well-aware of the numbering system used for the traditional bases, as seen below.
My question is how are the atoms in the size-expanded bases seen in xDNA and xRNA numbered?
This is a question of chemical nomenclature and the principle source for this is the IUPAC (IUBMB in case of biological molecules; but not in this case). You can find all hetero-ring nomenclature references on the IUPAC web site:
(I'm skipping the actual naming step)
Numbering is actually done according to http://www.acdlabs.com/iupac/nomenclature/79/r79_72.htm
That means, if you take the skeleton without hetero atoms, you start
Now, if there are more than one possibility due to symmetry, you chose the numbering that gives the heteros the smallest possible numbers (this means your purines would be actually wrongly numbered if they weren't an exception---see the List of retained names).
So, we have finally in dxC and dxT the start at the lower nitrogen, going anti-clockwise (the other N at 3, methyl at 6, ribose at 8), and in dxA and dxG similarly the start at the lower right N, going anti-clockwise with other Ns at 3, 6, and 8, and ribose at 8, as well.