Dear @mauellano
That a tricky one,
Its dependent on the Permalink structure of your pages, normally the way a language plugin works:
You set a 'home' language - it our case english so we reach a page with the permalink :
example.com/pageabout
However a language plugin will create a duplicate of each page and assign a language code to the permalink, so:
example.com/nl/pageabout
The language switch button makes all links clicked to have this /nl/ ( dutch ) placed in front of the standard permalink redirecting it to the correct language.
So my thoughts are you first need all the pages of a particular language to have a 'language code'.
Then you need a piece of jQuery, JS code added that first knows that your language button has been selected - probably via a simple cookie setup
Then if the browser knows that the button is pressed, it makes a links that are clicked to have your language code place into the permalink - redirecting you.
Its a tricky one but here are some references if they help you:
https://stackoverflow.com/questions/1458724/how-do-i-set-unset-a-cookie-with-jquery
https://stackoverflow.com/questions/20475504/onclick-change-url-using-jquery
https://wordpress.stackexchange.com/questions/89382/creating-custom-permalink-structure-for-languages
There is possibly a simpler option which is literally having two different website with the same domain - ( one being a subdomain )
Your main website is standard, then you create a subdomain e.g
nl.example.com
( you can create a subdomain with your domain/hosting provider )
And the link in the top corner just takes you to a clone of the website but with the different langauge:
https://premium.wpmudev.org/blog/multilingual-wordpress-site-multisite/
This is how i would go about it, if you don't want a plugin - it is the simpler option.
If this seems to complicated a task, you may wish to seek the Aid of a developer as this is an interesting one - there is unfortunately a good reason so many of us rely upon language plugins instead of doing the work manually
I hope this helps & apologies that the answer isn't as simple as may have been hoped
Best wishes and let me know, Have a great day :)
Sincerely
Richard