hello,
We process some tests directly on your website, and we weren't able to reproduce your issue, let's details our results.
We have add a product to the cart, we see the popup notification and choose "continue shopping"
=> No cookie question & the product is in the cart.
Now in order to progress and better understand your issue can you process this same test without being logged ? With several different browser ? And tell us which Browser did you use usually AND if you see differences from previous test.
Awaiting your returns to be able to progress on your issue.
Regards