If I was going to do this, I would probably go Paladin 20/Sorcerer 20 or Paladin 20/Bard 20. Either one of those combinations would just be silly powerful.
I understand that you don't want a caster so I would stick with Fighter 20/Barbarian 20 as I said before. This combination would be a ridiculous melee monster. Battle Master is always great, but I prefer the synergy Champion would give you with the Barbarian with the crits, extra fighting style, and regen 12hp at the start of your turn. You could take the GWF and Archery Fighting styles and not have a weakness. Or Dueling and go S&B if you REALLY want a silly high AC(22AC without shield, 24 with). I don't think I would go Polearm Master. I'd use a Greataxe and take GWM, Sentinel, Alert, and maybe Lucky and/or Mobile assuming you're going to have twice as many asi chances as normal.
I still think the Paladin is the strongest class and a Fighter 20/Paladin 20 would be extremely good too. You could ignore spells and just spam Divine Smite. All of the Pally's Auras are very strong. The Devotion Pally adds Charisma bonus to attacks so you could completely cancel out the -5 to hit from GWM assuming 20 Char. Oath of Ancients Pally + Champion Fighter would regain 20hp at the start of every turn. Your party needs at least 1 Paladin 20 IMO. Hell, you all should take Paladin 20 and choose a different Oath so everything would be covered haha. That party with be nigh unstoppable.
It just occurred to me how awesome Tempest Cleric 20/ Storm Sorcerer 20 would be............ lol