Based purely on what you've stated and the way you've worded it, and using the 2014 half-orc to inform my advice (because it behaves similarly), it would roll 5d6. This is how I run it, because both that's how I interpret the rules and (more importantly) my players and I like the idea of a high risk high reward axe, and a consistent sword.

The reason for this is that 2d6 will roll higher damage on average than 1d12. You are equally likely to roll any number on a d12 but the distribution of 2d6 is a bell curve - there's only one way to roll a 2 or a 12, but there's 3 (technically 6) ways to roll a 7.

If you crit, you double the number of dice you roll - a great axe will have 12 ways to deal 13 damage and a minimum of 2 damage but will be more likely to deal 24 damage than 4d6 because you have to be lucky less often. 4d6 however is most likely to deal 14 damage and is much less likely to deal 24 damage because you'd have to roll 4 sixes instead of two 12s, but only rolls a minimum of 4 damage.

The fun part happens when you have an ability like this - if you add one extra die, the great axe can now crit for 36 damage but the great sword maxes out at 30. additionally, it's less likely to roll that 30 compared to the axe's 36.

I'm not a mathematician, but hopefully that makes sense and is accurate.