From T2 to T3, you will have to go via T1. See
https://www.ifly.com/Manchester-Inte...t/terminal-map
The distance from T1 to T3 is shorter but is partially "outdoors", while T1 to T2 is "indoors" and has moving walkways.
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.