Let's say you have a website and you get a US trademark for its name in all of the relevant classes. What protection does this single-country trademark provide for a website?

I'm slightly confused because I asked a lawyer what would then happen if someone from, say, the UK subsequently registered an identical trademark for their company/website via the UK trademark body, would they have any legal right to demand the closure/alteration of our website?

He said that If I have no UK trademark rights and a third party obtains such rights then they may prevent me from selling my goods/services into the UK via any website.

Now I can see how that makes sense if your business sells a physical product - so, in this scenario you can sell your product anywhere in the US but you cannot ship it to the UK.

But if your product/service is merely your website's content (you make money from it through advertising/affiliate schemes, perhaps), what happens then? You have the US trademark (and own the domain), someone else has the UK trademark for the same name... does this mean that you will have to alter/take down the website, block it for UK visitors, etc.?

Anyone have any ideas what happens in this scenario?