Something of note...
In your profile page it says your location is Sarajevo and you also mentioned employment agencies. If you are going through international recruiting agencies you won't have much luck in games on any system. It is uncommon for studios to bring in foreign applicants. They may interview foreign people who are already moved locally, but generally they won't arrange for you to come to them.
It looks like there is only one studio in your area, although the list may be wrong.
You generally need to be local to get a game development job. You might need to move to a region with many studios, THEN look for a game development job.
I have to disagree based on my own experience.
I certainly don't know how common my situation was, but while I lived in Slovakia, I used to get a lot of phone / onsite interviews in UK from game companies, despite not actually having been employed by a proper huge game company (unless you count establishing your own indie company as a valid experience).
So, it's certainly possible. Of course, UK is not US, so only a select few companies were willing to participate in costs for the plane ticket and hotel (in which case it helped having a friend in London where I could stay for few days).
In the end, however, the resulting offers were always artificially lowered because I lacked the 'AAA' license game name (the good old argument of 'Your engine was rendering non-licensed characters ? Then it is not a proper engine! ').
Despite UK being more and more multicultural, you have to be prepared for incredibly ridiculous questions related to your country of origin. I would suspect that might be even worse in your case, as you're from Sarajevo. But, you might be lucky...
I can also confirm that networking during events like ECTS / GDC helped immensely. Once people have your business card, you will be contacted (even 6-12 months after the given GDC / ECTS event) when they are in the hiring phase. So, if you truly want to get into gamedev, this is a proven (albeit expensive) route.
Of course, I assume you already are listed with several UK gamedev agencies.
If you, however, need the Visa / work permit, then simply forget about it, unless you are already a proven top talent, in which case you would still be fighting against local top talent (which always wins, by definition).
Is Bosna&Hercegovina already a part of EU or not ? Sorry for not keeping up with latest EU inclusions - I cut myself off the news on Europe.
Also, note that about a year or two ago (could be more), the limitation present in original EU-founding countries on labour market (the one, where even if you are from a country within EU, you still need a work permit) has expired. Of course, if your country is not in EU, then it's of no help...