Somehow, I thought they would suggest this bodge.
Firstly two layers is not as strong as one thicker board.
Secondly, it is impossible to lay boards on top of existing in the opposite direction as the new ones will not cross the joists and only be supported and fixed to the existing flooring.
Thirdly, There will be extra weight on the floor which I don't think would have been allowed for.
Do not accept this bodge!
The studwork needs to be removed and the old 18mm removed and 22mm T&G MR flooring fixed to the joists.
Be careful that the staircase is ordered to suit a measured as is, storey height and not from the plans!