Hi,
I see the same £0.00 taxes on both FireFox and Chrome on my end.
I suppose you have some caching on FireFox preventing you from seeing the change, or you're logged in on FireFox and not on Chrome.
Taxes apply based on the address of the customer and the tax rules you created.
Before logging in / registering and providing an address, the system bases itself on the "main tax zone" setting of the HikaShop configuration instead of the user address.
So, from what I can see on your website, your explanations, and what I just wrote above, I think that there is a problem between the "main tax zone" you've configured and the tax rules that you've setup, which leads to HikaShop not displaying taxes for guest users.
Note also that if you have the "geolocation" plugin activated, it can override the "main tax zone" by the gelocalized zone of the user based on his IP address. So you need to take this into account if you're using that plugin.