You normally have to pay UK VAT on goods you export to EU customers who aren’t VAT registered, no tax for customers outside the EU. I think the exception is “digital goods” where an EU customer pays the VAT rate applicable in the country of purchase. Presumably this applies to your products.
You’re either going to have to show prices without tax and put a note that appropriate tax will be added after the customer enters their address, or, find out which country the customer lives in before they start to shop. WooCommerce has had geolocate for a while and this doesn’t generate much flak on this forum, so I understood it was working well for this purpose? Alternatively, get customers to register and login before they shop, though I accept this may put some customers off.
There’s a plugin to help with this process:
https://www.ads-software.com/plugins/woocommerce-eu-vat-compliance/
You may find other similar plugins on www.ads-software.com, woocommerce.com or codecanyon.
In my shop I ship only to the billing address. To ship to another address seems to carry some additional risk.
If the tax varies per customer, presumably you should enter prices without tax.
Your standard tax rates table will need a line for each EU country, and at the bottom of the table, an everywhere else line with * wildcards and 0% tax. This line will be ignored if the customer’s country is found higher up the table. The plugin mentioned should do this for you.