Speaking as someone who was a game programmer, it will make it harder.
The best programmer I’ve ever worked with didn’t have a CompSci degree. He had no degree. But he had been programming games professionally since he was 14.
When I met him, we were both in our twenties, but he had knowledge that was light-years beyond anyone else at the company, even people who had been programming for 30 years.
That was 20+ years ago. He recently told me the salary he made there. It was almost half of what I made. I assume it’s because I had a degree and he didn’t.
Time went on and that game studio went under. I got another job fairly quickly (but it was in a totally different part of the country) but he struggled. He showed the game studios the games he had worked on, he totally aced the technical interviews and even showed the interviewers where they were wrong on answers (which actually might’ve hurt him).
But he found it really hard to land another gig. Despite his experience and knowledge, other studios were reluctant to hire him because he didn’t have a degree. His résumé of games didn’t help; he was going up against new college grads who had CompSci degrees.
Eventually he was hired on with a studio, but only because a former co-worker of ours vouched for him. As soon as he was hired, he blew them away with his proficiency. He wrote a new networked Lua debugger for them in a few weeks that every other studio was scrambling for as soon as they saw it.
He was—is—the greatest programmer I’ve ever worked with. But even back in the late 1990s and early 2000s he had trouble landing gigs. The CompSci degree speaks volumes.
Today there’s a huge oversupply of eager candidates trying to get into game development. Many aspiring game programmers have CompSci (or similar) degrees. Though you might have some nice projects to show, without that degree, you might not even get the chance to show them.
In industries outside of game development, you might have a better shot. Software engineers are in high demand right now and there aren’t always a huge number of candidates. Personal projects will work here too.
I don’t want to be a wet blanket. You can still apply for game development (programming) jobs. Failing an interview or two won’t hurt you in any way. But know the competition is stiff.
Do have some game projects to show them. Be prepared to explain how you overcame certain challenges. Be prepared to explain how certain things in your game work. If you get an interview, be prepared for tough technical questions.