Not yet! (if anybody here had done so we would certainly have updated the wikipedia pages).
The methodology would, I expect, be along similar lines to the original Sudoku 9x9 case. See
here for a relatively recent discussion with some useful links.
The Burnside's Lemma approach would seem to be only way forward, since it does not require enumeration of all the grids, but even with this method the numbers are still daunting. This was the method used to count the ED Sudoku 9x9 grids.
To begin with we would need to identify equivalence (conjugacy) classes. The only software for this is a group-theory package like GAP. The number of VPT's (Validity Preserving Transformations) involved, is, I believe, 2 x (12 ^ 10) =
123,834,728,448 for 16x16 Sudoku, compared with just 2 x 6^8 = 3,359,232 for 9x9 Sudoku.
With a group of that size, it's highly unlikely that any desktop PC is going to be able to accomplish even this first step! Some serious architecture would need to be thrown at the problem ...