I also think you will be OK to check luggage through to the second flight, just have all the details handy, particularly the flight number and PNR.

For the transfer, there are 2 places you could end up if coming in on from LCY, and what it boils down to is that you best transfer landside. It is very clearly sign posted, it's a bit of trek (go up to departures first) and you end up passing outside briefly, but there is a corridor by the roadside section of the terminal building. 10 minutes to do that, 5 minutes for security in the modern T2 building.

You could use - in the mornings - the Flight Connection route by baggage collection, but particularly if you were landed at the 200 gates it wouldn't make much sense. T1 and T2 are effectively one building once airside, but landing into T1 pushes you to landside anyway due to the location of passport control (which isn't the case if landing at T2).