I am trying to sign in to my apple tv+ account on a Windows Laptop.
You have two options:
I am able to watch titles on my phone app but I am unable to sign in on my laptop. There is no blue sign in button and when I click on the icon (person in circle), nothing happens.
The web access at tv.apple.com would have a blue Sign In button in the top-right corner, but inside the webpage (the black bar). When the browser remembers your last use, it may sign you in automatically (through cookies). When signed in, instead of a Sign In button, you would see a generic account icon, or your Apple ID personal image, if configured. Don’t confuse it with the account button in the toolbar that some browsers may show, as that is about the browser, not the visited webpage.
Apple recommends the Chrome or Firefox browser.

Could this be related to the fact that I bought my first apple product in the UK (and therefore my apple ID is associated with the UK) but I am currently in the Netherlands?
Possibly, but I don’t think so. Enough users have reported being able to access Apple TV+ from beyond their national border. Officially, Apple only offers services for your “home country”, so they won’t have to make additional efforts for international use. Possibly the exact originating and visiting country matter. YMMV.