On separate tickets, SQ is more likely to interline luggage to LH, almost certainly not to BA. If your luggage is interlined, or you don't have any, you can transfer airside by bus without going through UK immigration.
I went ahead and booked BA because the flight times to Krakow were more convenient than LH. Regarding your comment above, if my flight originates on United Airlines AUS-IAH, would it be possible for UA to interline my luggage all the way to MAN?