I have booked the MAN-LHR-ICN itinerary as in the original post, via the KE web site.
Travelling in KE Prestige (Business) class.
When I check in at the BA desk at MAN, will I get all boarding cards?
Or will I need to go to a KE desk in LHR T4, for document checks and a KE boarding card?

My experience with BA/AA itineraries is that AA rips up the BA-issued boarding card and reissues on AA stock. And vice versa.
And they are both in the OneWorld alliance...