TL;DR: Check if a browser extension is blocking your request when you click Check Now
While playing more with jenkins, I've encountered an error while requesting Check Now for plugin updates: it seemed that my request got ignored. I've tried different URLs (with and without HTTPS, stable or not) with the same result: my request seemed to be ignored. I've assumed first there's a proxy error or some limitation on the server side. The logs however didn't reveal anything. Then, I've checked the browser side and I've found that my Privacy Badger plugin blocked the request to the url (and subsequently to mirrors).
My solution was to disable the Privacy Badger for my jenkins install.