Ask HN: Is a SaaS provider expected to have a trademark in every country?

I'm wondering how international law is affecting SaaS Providers. For example, let's say Notion.so has a trademark in the US for the name "Notion". Then someone builds, theoretically, a similar service in the UK, and trademarks "Notion" there. Can the UK entity prevent "Notion Labs" (that doesn't have an office in the UK) to sell to customers in the UK? In a broader perspective, are SaaS providers, that inherently have customers all over the world, expected to register a trademark in all countries or only the countries they are supposed to have offices in?