So the pick axe movement is not clearly explained, so I found tinkering with the game settings helped.

So from the Main Menu of the Game. Options > Accessibility or if you are in the game already, Pause > Accessibility.

Toggle Climbing Axe - If you turn this on, you do not have to hold down the mouse buttons or trigger buttons if on controller. It does however mean you have Click once to attache, then de-attache.

Start Climbing With One Button - If you turn this on, you do not have to press the Left and Right Mouse Buttons or Both triggers at the same time, you just have to click one button.

I found the toggled Climbing Axe On was much easier to control on a mouse, and lessens any accidents from miss clicks or taking any pressure off the buttons.
The Climbing With One Button, I found when using a mouse you had to be very precise with both buttons to get a good hold when jumping to climbable structures.

Movements:
With what ever you choose mouse or controller. The game explains that if you need to jump from one climbing structure to the right for example, you need to have your Left Axe in the structure, and your Right free with you leaning to the right and then press Space Bar or the respective Jump Button on Controller to make the jumps then you do the attaching method to hang onto the structure and not fall and fail the jump.

But you can also use S then the Space Bar if you need to jump backwards like on Chapter 6 in the cave with the ice.
That was not explained anywhere in the game, but I found out with trial and error that is what I needed to do.

I hope this helps anyone who is a bit frustrated with the Axe mechanics.