I do the LHR-KGX journey on the Picc quite a lot (though from KGX I go only to Cambridge, not the frozen North). According to https://tfl.gov.uk , it takes 12 minutes to walk from Terminal 2 to the shared T123 underground station. In my experience the TFL walking times are generous, but I suspect it's around 10 minutes.
According to the same site, the Piccadilly T123-KGX service runs every 5 minutes (successive Saturday afternoon departures are listed at 1439, 1444, and 1449). I wouldn't rely on those exact times, but it does mean the service is frequent, so rocking up and taking the next tube is a reasonable thing to do.
Don't forget you will need to allow time to deplane and clear customs and immigration. Even without luggage to collect, this can take some time; I'd allow an hour to be safe (this will be decreased by being an EU citizen, but increased if you don't have an e-passport). So we're looking at something like 60mn deplane + 10mn walk + up to 5mn wait for tube + 55mn on the tube, for a total of 130mn.
I think you would be very foolish to allow less than two hours between touchdown and train-departs-KGX. 2h15m might be all right, though you may end up joining the sprinters through KGX, and 2h30m is a fairly safe bet which may give you time for a pastry at KGX (Patisserie Valerie is on the upstairs level, nom nom). Be sure to be at the rear of the Piccadilly line train to get off at the right place at KGX for fastest access to street level. Have your tube ticket in your right hand to avoid delay at the barriers.
The return journey is even harder to time, because you don't tell us how long before departure your airline requires you to arrive. You don't need to allow the deplaning time, so something like 70 minutes from KGX to T123 is probably optimal. If you plan to arrive KGX an hour and a half before you need to be at check-in, it will likely be enough.
The last caveat is to check service online before you do this. Weekends are when TFL performs most of its maintenance, and Crossrail is still playing hob with London journey times. You're planning to shave things fine, so a 20 minute delay at Acton Town will kill you.