Language of the browser does not directly matter. It’s the language preference for the web sites which matter, though if you install eg. Slovak language browser it might by default include the Slovak language in your language preferences. I personally have Firefox UI in English but my language preferences for web sites also include my local language along with English.
In my earlier reply I already described the process to change the language preference for the web sites in Firefox. Similar option should be present in the other browsers, too. I don’t personally use Chrome so I can’t exactly tell where such preferences are. From what I remember you can add multiple languages and order them in chrome://settings/languages with each language having option to make it also the UI language and optional translation target. For this issue the English (UK) should be higher than English (US) or plain English.
Whole protocol level language negatition was in theory good idea but in practice it never really worked that well so most sites (for a reason) don’t use it. Especially to determine regions…