I believe that as a US citizen I can enter the UK or the EU (until 2021) without a visa for tourism or business purposes for up to 6 months. (Correct me if I'm wrong.)

I'm an independent consultant with clients in the USA. While traveling - for leisure! - I will from time to time also do work for these clients - not just checking email, but also writing/reviewing documents, programming, etc.

This is presumably allowed. (If not, why not?)

My question is: Do I need to mention any of this if I somehow get stopped on entrance and asked what the purpose of my visit is? I'd like to know if the one-word answer "tourism" is sufficient in this case.