I tested on browserstack to test with the same environment as you. I don't always have the error, but sometimes : hidden link
Could you try to set the address in the checkout page please ? It should be displayed a loader, but it loads to the infinity. And if you go back to the cart page you have the timeout too.
After more tests on my side, I confirm that you need to have a address set on the checkout page to have the error. The timeout seems to be triggered by some methods to get shipping price and taxes. If we try to fix an issue, an another is triggered by an another plugin. The first timeout we have seems to be triggered by the gift card, we fix it, and then we have the problem with the class 'ups_carrier', and it may be continue if we fix this. The source of the problem is not from the other plugins but from WCML in my opinion, but I may be wrong...
Could you make a last test on the staging to relicate the error on your side please ? Make a first order (with "Virement bancaire" as payment method) with your account. A default address wil be added to your account. Then make a new cart with the product, and you should always have the timeout error.
I can not desactivated all plugins on the staging, I need to have a functionnal site for others tests.
I've rechecked and I don't think this is related to WCML as I'm able to replicate the crash that you encountered as shown in the attached screenshot. The issue happens even after I've deactivated WCML.