There's not much in it between going east via the M67 and then south on the M1 (as Google recommends here -
https://www.google.co.uk/maps/dir/Ma...701,-2.0797385, or following Ray's suggestion. Personally, I'd take the former - it's a nicer drive, and less chance of a minor accident having a major impact on your journey time - the M6 south between Manchester and Stafford is just one permanent set of road works, which are typically worse at the weekend, so if you enjoy the pleasure of driving at all, then I'd avoid that.
Given that you're aiming to be in Nottingham for 6.30pm on Tuesday, I'd definitely favour the M67/M1 route - the M6 south from 3pm can be at a standstill for over an hour - on my last two trips south at that time Google (correctly) diverted me onto A roads, and it took over 2 and 1/2 hours just to get to Birmingham.