Hello,
The browser console indicate some errors
hikashop.js:25 Uncaught TypeError: Cannot read property 'classList' of null
at Object.addClass (hikashop.js?v=401:25)
at Object.setLoading (checkout.js:143)
at Object.submitBlock (checkout.js:93)
at Object.window.checkout.submitCart (checkout:269)
at HTMLAnchorElement.onclick (checkout:213)
And the reason of that error is related of the missing "btn" :
btn = d.getElementById('hikabtn_checkout_next')
So it means that your template overrides are changing elements and do not display the "checkout next" button as it should be.
Javascript cannot find.access to that button to interact with it ; which generate JS errors and the reason why nothing happens when you try to perform an action in the checkout.
Regards,