At street level Pancras Road is a barrier between the 2 stations. This is a busy road especially for taxis as it serves both drop off and pick up for Kings Cross plus drop off for St Pancras.
The Kings Cross St Pancras Underground station has 2 walkways under Pancras Road, one near Euston Road and one near Kings Boulevard. Both of these are outside the barrier line.
Kings Cross has three barrier lines, platforms 0-8, platforms 9/10 and the archway from the mezzanine level to the footbridge, with free movement between each barrier line and all platforms, though some are a long walk.
St Pancras (excluding Eurostar) also has 3 barrier lines, one each for East Midlands platforms 1-4, South Eastern platforms 11-13 and Thameslink platforms A/B, but these are all completely separate and it is not possible to move between each group on the platform side of the barrier lines.
As far as I'm aware there are no Cluedo style secret passages.
Has a train ever called at both in succession, perhaps in the last days of KX Thameslink and the first of StP Thameslink?
I would be very very surprised if this had ever happened. It is almost impossible to envisage any circumstances where this could be done but going to Finsbury Park or Kentish Town could not.
Kings Cross Thameslink closed and St Pancras International Low Level opened in 2007, and the Canal Tunnel opened in 2018, so it is a definite no for Kings Cross in the last days of Kings Cross Thameslink and the first of St Pancras International Low Level.
I used Kings Cross Thameslink on the last Friday before it closed and St Pancras International Low Level on the first Monday it opened; it was a very smooth transition.