They are breaking their journey if they:
"leave a Train Company’s or Rail Service Company’s stations after you start your journey other than:
(i) to join a train at another station"
http://www.nationalrail.co.uk/system/galleries/download/misc/NRCOC.pdf
Reffering to Appendix A, we find that a " “Rail Service Company” means Network Rail, any person (other than a Train
Company) who operates a station and any person who hires rolling stock to a Train
Company, but does not include London Underground", so when the passenger leaves Piccadilly Circus Station they are not leaving a Rail Service Company's station (as said station is owned and managed by LU.
"“Train Company” means a company operating passenger railway services which
is required to apply these Conditions to its tickets under a condition of the
Passenger Licence granted to it by the Office of Rail Regulation. A list of these
companies can be found in Appendix C. “Train Companies” means all or more
than one of these Companies;" London Underground does not appear in Appendix C, so no break of journey occurs at Piccadilly Circus.
Therefore, it is down to the reasoning for leaving Kings Cross station. If the intention prior to leaving the station is to do some shopping on the way to Paddington, then quite clearly the passenger is "leaving a Train Company’s or Rail Service Company’s
stations after you start their journey to join a train at another station and do some shopping" (my rephrasing from NRCoC), therefore they are breaking the journey, if they decide having left Kings Cross BR station, the position is less clear.