Is $Ω(n\log ⁡n)$ the lower-bound for *all* sorting algorithms or *just comparison-based* sorting algorithms?
O($n$ $log$ $n$) lower bound complexity assumes the length of each key bit is constant.  If not, then in the worst case you must compare nearly all bits, and multiply the complexity by that length and get O($kn$ $log$ $n$)

In contrast, some non comparison sorts with arbitrary key lengths can be done in worst case O($kn$).

EDIT: To add more detail:

For Counting Sort, the worst case and average case are O($n+r$) where $r$ is the range of keys (max - min).  

For LSD Radix sort the worst case and average case are O($n+k/d$) where $k$ is the key length and $d$ is the digit size.

You mention that there is no limit on key bit length.  It would seem that LDS Radix has better worst case performance:  $O(k(n+ln\ n))$ > $O(n+k/d$), putting aside other factors such as space.